diff --git a/src/neo4j/_async/io/_bolt.py b/src/neo4j/_async/io/_bolt.py
index 21f87ff74..4c9ebfbb9 100644
--- a/src/neo4j/_async/io/_bolt.py
+++ b/src/neo4j/_async/io/_bolt.py
@@ -28,6 +28,7 @@
 from ..._codec.hydration import v1 as hydration_v1
 from ..._codec.packstream import v1 as packstream_v1
 from ..._conf import PoolConfig
+from ..._deadline import Deadline
 from ..._exceptions import (
     BoltError,
     BoltHandshakeError,
@@ -289,17 +290,21 @@ def get_handshake(cls):
         return b"".join(version.to_bytes() for version in offered_versions).ljust(16, b"\x00")
 
     @classmethod
-    async def ping(cls, address, *, timeout=None, pool_config=None):
+    async def ping(cls, address, *, deadline=None, pool_config=None):
         """ Attempt to establish a Bolt connection, returning the
         agreed Bolt protocol version if successful.
         """
         if pool_config is None:
             pool_config = PoolConfig()
+        if deadline is None:
+            deadline = Deadline(None)
+
         try:
             s, protocol_version, handshake, data = \
                 await AsyncBoltSocket.connect(
                     address,
-                    timeout=timeout,
+                    tcp_timeout=pool_config.connection_timeout,
+                    deadline=deadline,
                     custom_resolver=pool_config.resolver,
                     ssl_context=pool_config.get_ssl_context(),
                     keep_alive=pool_config.keep_alive,
@@ -313,14 +318,14 @@ async def ping(cls, address, *, timeout=None, pool_config=None):
     # [bolt-version-bump] search tag when changing bolt version support
     @classmethod
     async def open(
-        cls, address, *, auth=None, timeout=None, routing_context=None,
+        cls, address, *, auth=None, deadline=None, routing_context=None,
         pool_config=None
     ):
         """Open a new Bolt connection to a given server address.
 
         :param address:
         :param auth:
-        :param timeout: the connection timeout in seconds
+        :param deadline: how long to wait for the connection to be established
         :param routing_context: dict containing routing context
         :param pool_config:
 
@@ -330,26 +335,17 @@ async def open(
             raised if the Bolt Protocol can not negotiate a protocol version.
         :raise ServiceUnavailable: raised if there was a connection issue.
         """
-        def time_remaining():
-            if timeout is None:
-                return None
-            t = timeout - (perf_counter() - t0)
-            return t if t > 0 else 0
 
-        t0 = perf_counter()
         if pool_config is None:
             pool_config = PoolConfig()
+        if deadline is None:
+            deadline = Deadline(None)
 
-        socket_connection_timeout = pool_config.connection_timeout
-        if socket_connection_timeout is None:
-            socket_connection_timeout = time_remaining()
-        elif timeout is not None:
-            socket_connection_timeout = min(pool_config.connection_timeout,
-                                            time_remaining())
         s, protocol_version, handshake, data = \
             await AsyncBoltSocket.connect(
                 address,
-                timeout=socket_connection_timeout,
+                tcp_timeout=pool_config.connection_timeout,
+                deadline=deadline,
                 custom_resolver=pool_config.resolver,
                 ssl_context=pool_config.get_ssl_context(),
                 keep_alive=pool_config.keep_alive,
@@ -410,7 +406,7 @@ def time_remaining():
         )
 
         try:
-            connection.socket.set_deadline(time_remaining())
+            connection.socket.set_deadline(deadline)
             try:
                 await connection.hello()
             finally:
diff --git a/src/neo4j/_async/io/_pool.py b/src/neo4j/_async/io/_pool.py
index 8376d5610..9058bb5a6 100644
--- a/src/neo4j/_async/io/_pool.py
+++ b/src/neo4j/_async/io/_pool.py
@@ -135,9 +135,7 @@ async def connection_creator():
             released_reservation = False
             try:
                 try:
-                    connection = await self.opener(
-                        address, deadline.to_timeout()
-                    )
+                    connection = await self.opener(address, deadline)
                 except ServiceUnavailable:
                     await self.deactivate(address)
                     raise
@@ -382,9 +380,9 @@ def open(cls, address, *, auth, pool_config, workspace_config):
         :returns: BoltPool
         """
 
-        async def opener(addr, timeout):
+        async def opener(addr, deadline):
             return await AsyncBolt.open(
-                addr, auth=auth, timeout=timeout, routing_context=None,
+                addr, auth=auth, deadline=deadline, routing_context=None,
                 pool_config=pool_config
             )
 
@@ -437,9 +435,9 @@ def open(cls, *addresses, auth, pool_config, workspace_config,
             raise ConfigurationError("The key 'address' is reserved for routing context.")
         routing_context["address"] = str(address)
 
-        async def opener(addr, timeout):
+        async def opener(addr, deadline):
             return await AsyncBolt.open(
-                addr, auth=auth, timeout=timeout,
+                addr, auth=auth, deadline=deadline,
                 routing_context=routing_context, pool_config=pool_config
             )
 
diff --git a/src/neo4j/_async_compat/network/_bolt_socket.py b/src/neo4j/_async_compat/network/_bolt_socket.py
index 18dec61fe..507d080a2 100644
--- a/src/neo4j/_async_compat/network/_bolt_socket.py
+++ b/src/neo4j/_async_compat/network/_bolt_socket.py
@@ -272,11 +272,11 @@ async def _connect_secure(cls, resolved_address, timeout, keep_alive, ssl):
                 ) from error
             raise
 
-    async def _handshake(self, resolved_address):
+    async def _handshake(self, resolved_address, deadline):
         """
 
-        :param s: Socket
         :param resolved_address:
+        :param deadline: Deadline for handshake
 
         :returns: (socket, version, client_handshake, server_response_data)
         """
@@ -296,47 +296,52 @@ async def _handshake(self, resolved_address):
         log.debug("[#%04X]  C: <HANDSHAKE> %s %s %s %s", local_port,
                   *supported_versions)
 
-        data = self.Bolt.MAGIC_PREAMBLE + self.Bolt.get_handshake()
-        await self.sendall(data)
+        request = self.Bolt.MAGIC_PREAMBLE + self.Bolt.get_handshake()
 
         # Handle the handshake response
         original_timeout = self.gettimeout()
-        if original_timeout is not None:
-            self.settimeout(original_timeout + 1)
+        self.settimeout(deadline.to_timeout())
         try:
-            data = await self.recv(4)
+            await self.sendall(request)
+            response = await self.recv(4)
         except OSError as exc:
             raise ServiceUnavailable(
-                "Failed to read any data from server {!r} "
-                "after connected".format(resolved_address)) from exc
+                f"Failed to read any data from server {resolved_address!r} "
+                f"after connected (deadline {deadline})"
+            ) from exc
         finally:
             self.settimeout(original_timeout)
-        data_size = len(data)
+        data_size = len(response)
         if data_size == 0:
             # If no data is returned after a successful select
             # response, the server has closed the connection
             log.debug("[#%04X]  S: <CLOSE>", local_port)
             await self.close()
             raise ServiceUnavailable(
-                "Connection to {address} closed without handshake response".format(
-                    address=resolved_address))
+                f"Connection to {resolved_address} closed without handshake "
+                "response"
+            )
         if data_size != 4:
             # Some garbled data has been received
             log.debug("[#%04X]  S: @*#!", local_port)
             await self.close()
             raise BoltProtocolError(
-                "Expected four byte Bolt handshake response from %r, received %r instead; check for incorrect port number" % (
-                resolved_address, data), address=resolved_address)
-        elif data == b"HTTP":
+                "Expected four byte Bolt handshake response from "
+                f"{resolved_address!r}, received {response!r} instead; "
+                "check for incorrect port number"
+                , address=resolved_address
+            )
+        elif response == b"HTTP":
             log.debug("[#%04X]  S: <CLOSE>", local_port)
             await self.close()
             raise ServiceUnavailable(
-                "Cannot to connect to Bolt service on {!r} "
-                "(looks like HTTP)".format(resolved_address))
-        agreed_version = data[-1], data[-2]
+                f"Cannot to connect to Bolt service on {resolved_address!r} "
+                "(looks like HTTP)"
+            )
+        agreed_version = response[-1], response[-2]
         log.debug("[#%04X]  S: <HANDSHAKE> 0x%06X%02X", local_port,
                   agreed_version[1], agreed_version[0])
-        return self, agreed_version, handshake, data
+        return self, agreed_version, handshake, response
 
     @classmethod
     async def close_socket(cls, socket_):
@@ -356,8 +361,8 @@ async def close_socket(cls, socket_):
                 pass
 
     @classmethod
-    async def connect(cls, address, *, timeout, custom_resolver, ssl_context,
-                      keep_alive):
+    async def connect(cls, address, *, tcp_timeout, deadline,
+                      custom_resolver, ssl_context, keep_alive):
         """ Connect and perform a handshake and return a valid Connection object,
         assuming a protocol version can be agreed.
         """
@@ -371,12 +376,18 @@ async def connect(cls, address, *, timeout, custom_resolver, ssl_context,
             addressing.Address(address), resolver=custom_resolver
         )
         async for resolved_address in resolved_addresses:
+            deadline_timeout = deadline.to_timeout()
+            if (
+                deadline_timeout is not None
+                and deadline_timeout <= tcp_timeout
+            ):
+                tcp_timeout = deadline_timeout
             s = None
             try:
                 s = await cls._connect_secure(
-                    resolved_address, timeout, keep_alive, ssl_context
+                    resolved_address, tcp_timeout, keep_alive, ssl_context
                 )
-                return await s._handshake(resolved_address)
+                return await s._handshake(resolved_address, deadline)
             except (BoltError, DriverError, OSError) as error:
                 try:
                     local_port = s.getsockname()[1]
@@ -560,11 +571,12 @@ def _secure(cls, s, host, ssl_context):
         return s
 
     @classmethod
-    def _handshake(cls, s, resolved_address):
+    def _handshake(cls, s, resolved_address, deadline):
         """
 
         :param s: Socket
         :param resolved_address:
+        :param deadline:
 
         :returns: (socket, version, client_handshake, server_response_data)
         """
@@ -584,46 +596,52 @@ def _handshake(cls, s, resolved_address):
         log.debug("[#%04X]  C: <HANDSHAKE> %s %s %s %s", local_port,
                   *supported_versions)
 
-        data = cls.Bolt.MAGIC_PREAMBLE + cls.Bolt.get_handshake()
-        s.sendall(data)
+        request = cls.Bolt.MAGIC_PREAMBLE + cls.Bolt.get_handshake()
 
         # Handle the handshake response
-        ready_to_read = False
-        with selectors.DefaultSelector() as selector:
-            selector.register(s, selectors.EVENT_READ)
-            selector.select(1)
+        original_timeout = s.gettimeout()
+        s.settimeout(deadline.to_timeout())
         try:
-            data = s.recv(4)
+            s.sendall(request)
+            response = s.recv(4)
         except OSError as exc:
             raise ServiceUnavailable(
-                "Failed to read any data from server {!r} "
-                "after connected".format(resolved_address)) from exc
-        data_size = len(data)
+                f"Failed to read any data from server {resolved_address!r} "
+                f"after connected (deadline {deadline})"
+            ) from exc
+        finally:
+            s.settimeout(original_timeout)
+        data_size = len(response)
         if data_size == 0:
             # If no data is returned after a successful select
             # response, the server has closed the connection
             log.debug("[#%04X]  S: <CLOSE>", local_port)
             cls.close_socket(s)
             raise ServiceUnavailable(
-                "Connection to {address} closed without handshake response".format(
-                    address=resolved_address))
+                f"Connection to {resolved_address} closed without handshake "
+                "response"
+            )
         if data_size != 4:
             # Some garbled data has been received
             log.debug("[#%04X]  S: @*#!", local_port)
             cls.close_socket(s)
             raise BoltProtocolError(
-                "Expected four byte Bolt handshake response from %r, received %r instead; check for incorrect port number" % (
-                resolved_address, data), address=resolved_address)
-        elif data == b"HTTP":
+                "Expected four byte Bolt handshake response from "
+                f"{resolved_address!r}, received {response!r} instead; "
+                "check for incorrect port number"
+                , address=resolved_address
+            )
+        elif response == b"HTTP":
             log.debug("[#%04X]  S: <CLOSE>", local_port)
             cls.close_socket(s)
             raise ServiceUnavailable(
-                "Cannot to connect to Bolt service on {!r} "
-                "(looks like HTTP)".format(resolved_address))
-        agreed_version = data[-1], data[-2]
+                f"Cannot to connect to Bolt service on {resolved_address!r} "
+                "(looks like HTTP)"
+            )
+        agreed_version = response[-1], response[-2]
         log.debug("[#%04X]  S: <HANDSHAKE> 0x%06X%02X", local_port,
                   agreed_version[1], agreed_version[0])
-        return cls(s), agreed_version, handshake, data
+        return cls(s), agreed_version, handshake, response
 
     @classmethod
     def close_socket(cls, socket_):
@@ -639,8 +657,8 @@ def close_socket(cls, socket_):
             pass
 
     @classmethod
-    def connect(cls, address, *, timeout, custom_resolver, ssl_context,
-                keep_alive):
+    def connect(cls, address, *, tcp_timeout, deadline, custom_resolver,
+                ssl_context, keep_alive):
         """ Connect and perform a handshake and return a valid Connection object,
         assuming a protocol version can be agreed.
         """
@@ -653,12 +671,19 @@ def connect(cls, address, *, timeout, custom_resolver, ssl_context,
             addressing.Address(address), resolver=custom_resolver
         )
         for resolved_address in resolved_addresses:
+            deadline_timeout = deadline.to_timeout()
+            if (
+                deadline_timeout is not None
+                and deadline_timeout <= tcp_timeout
+            ):
+                tcp_timeout = deadline_timeout
             s = None
             try:
-                s = BoltSocket._connect(resolved_address, timeout, keep_alive)
+                s = BoltSocket._connect(resolved_address, tcp_timeout,
+                                        keep_alive)
                 s = BoltSocket._secure(s, resolved_address._host_name,
                                        ssl_context)
-                return BoltSocket._handshake(s, resolved_address)
+                return BoltSocket._handshake(s, resolved_address, deadline)
             except (BoltError, DriverError, OSError) as error:
                 try:
                     local_port = s.getsockname()[1]
diff --git a/src/neo4j/_deadline.py b/src/neo4j/_deadline.py
index cfcc9035d..560e55a54 100644
--- a/src/neo4j/_deadline.py
+++ b/src/neo4j/_deadline.py
@@ -72,6 +72,9 @@ def from_timeout_or_deadline(cls, timeout):
             return timeout
         return cls(timeout)
 
+    def __str__(self):
+        return f"Deadline(timeout={self._original_timeout})"
+
 
 merge_deadlines = min
 
diff --git a/src/neo4j/_sync/io/_bolt.py b/src/neo4j/_sync/io/_bolt.py
index f1d7f1528..09f832e20 100644
--- a/src/neo4j/_sync/io/_bolt.py
+++ b/src/neo4j/_sync/io/_bolt.py
@@ -28,6 +28,7 @@
 from ..._codec.hydration import v1 as hydration_v1
 from ..._codec.packstream import v1 as packstream_v1
 from ..._conf import PoolConfig
+from ..._deadline import Deadline
 from ..._exceptions import (
     BoltError,
     BoltHandshakeError,
@@ -289,17 +290,21 @@ def get_handshake(cls):
         return b"".join(version.to_bytes() for version in offered_versions).ljust(16, b"\x00")
 
     @classmethod
-    def ping(cls, address, *, timeout=None, pool_config=None):
+    def ping(cls, address, *, deadline=None, pool_config=None):
         """ Attempt to establish a Bolt connection, returning the
         agreed Bolt protocol version if successful.
         """
         if pool_config is None:
             pool_config = PoolConfig()
+        if deadline is None:
+            deadline = Deadline(None)
+
         try:
             s, protocol_version, handshake, data = \
                 BoltSocket.connect(
                     address,
-                    timeout=timeout,
+                    tcp_timeout=pool_config.connection_timeout,
+                    deadline=deadline,
                     custom_resolver=pool_config.resolver,
                     ssl_context=pool_config.get_ssl_context(),
                     keep_alive=pool_config.keep_alive,
@@ -313,14 +318,14 @@ def ping(cls, address, *, timeout=None, pool_config=None):
     # [bolt-version-bump] search tag when changing bolt version support
     @classmethod
     def open(
-        cls, address, *, auth=None, timeout=None, routing_context=None,
+        cls, address, *, auth=None, deadline=None, routing_context=None,
         pool_config=None
     ):
         """Open a new Bolt connection to a given server address.
 
         :param address:
         :param auth:
-        :param timeout: the connection timeout in seconds
+        :param deadline: how long to wait for the connection to be established
         :param routing_context: dict containing routing context
         :param pool_config:
 
@@ -330,26 +335,17 @@ def open(
             raised if the Bolt Protocol can not negotiate a protocol version.
         :raise ServiceUnavailable: raised if there was a connection issue.
         """
-        def time_remaining():
-            if timeout is None:
-                return None
-            t = timeout - (perf_counter() - t0)
-            return t if t > 0 else 0
 
-        t0 = perf_counter()
         if pool_config is None:
             pool_config = PoolConfig()
+        if deadline is None:
+            deadline = Deadline(None)
 
-        socket_connection_timeout = pool_config.connection_timeout
-        if socket_connection_timeout is None:
-            socket_connection_timeout = time_remaining()
-        elif timeout is not None:
-            socket_connection_timeout = min(pool_config.connection_timeout,
-                                            time_remaining())
         s, protocol_version, handshake, data = \
             BoltSocket.connect(
                 address,
-                timeout=socket_connection_timeout,
+                tcp_timeout=pool_config.connection_timeout,
+                deadline=deadline,
                 custom_resolver=pool_config.resolver,
                 ssl_context=pool_config.get_ssl_context(),
                 keep_alive=pool_config.keep_alive,
@@ -410,7 +406,7 @@ def time_remaining():
         )
 
         try:
-            connection.socket.set_deadline(time_remaining())
+            connection.socket.set_deadline(deadline)
             try:
                 connection.hello()
             finally:
diff --git a/src/neo4j/_sync/io/_pool.py b/src/neo4j/_sync/io/_pool.py
index 9fe584284..229cc8c91 100644
--- a/src/neo4j/_sync/io/_pool.py
+++ b/src/neo4j/_sync/io/_pool.py
@@ -135,9 +135,7 @@ def connection_creator():
             released_reservation = False
             try:
                 try:
-                    connection = self.opener(
-                        address, deadline.to_timeout()
-                    )
+                    connection = self.opener(address, deadline)
                 except ServiceUnavailable:
                     self.deactivate(address)
                     raise
@@ -382,9 +380,9 @@ def open(cls, address, *, auth, pool_config, workspace_config):
         :returns: BoltPool
         """
 
-        def opener(addr, timeout):
+        def opener(addr, deadline):
             return Bolt.open(
-                addr, auth=auth, timeout=timeout, routing_context=None,
+                addr, auth=auth, deadline=deadline, routing_context=None,
                 pool_config=pool_config
             )
 
@@ -437,9 +435,9 @@ def open(cls, *addresses, auth, pool_config, workspace_config,
             raise ConfigurationError("The key 'address' is reserved for routing context.")
         routing_context["address"] = str(address)
 
-        def opener(addr, timeout):
+        def opener(addr, deadline):
             return Bolt.open(
-                addr, auth=auth, timeout=timeout,
+                addr, auth=auth, deadline=deadline,
                 routing_context=routing_context, pool_config=pool_config
             )
 
diff --git a/tests/unit/async_/io/test_direct.py b/tests/unit/async_/io/test_direct.py
index 7729036ea..541fff6fb 100644
--- a/tests/unit/async_/io/test_direct.py
+++ b/tests/unit/async_/io/test_direct.py
@@ -102,14 +102,17 @@ async def acquire(
 @mark_async_test
 async def test_bolt_connection_open():
     with pytest.raises(ServiceUnavailable):
-        await AsyncBolt.open(("localhost", 9999), auth=("test", "test"))
+        await AsyncBolt.open(
+            ("localhost", 9999), auth=("test", "test")
+        )
 
 
 @mark_async_test
 async def test_bolt_connection_open_timeout():
     with pytest.raises(ServiceUnavailable):
-        await AsyncBolt.open(("localhost", 9999), auth=("test", "test"),
-                             timeout=1)
+        await AsyncBolt.open(
+            ("localhost", 9999), auth=("test", "test"), deadline=Deadline(1)
+        )
 
 
 @mark_async_test
@@ -120,7 +123,9 @@ async def test_bolt_connection_ping():
 
 @mark_async_test
 async def test_bolt_connection_ping_timeout():
-    protocol_version = await AsyncBolt.ping(("localhost", 9999), timeout=1)
+    protocol_version = await AsyncBolt.ping(
+        ("localhost", 9999), deadline=Deadline(1)
+    )
     assert protocol_version is None
 
 
diff --git a/tests/unit/sync/io/test_direct.py b/tests/unit/sync/io/test_direct.py
index bd5dbe70d..45a901b8e 100644
--- a/tests/unit/sync/io/test_direct.py
+++ b/tests/unit/sync/io/test_direct.py
@@ -102,14 +102,17 @@ def acquire(
 @mark_sync_test
 def test_bolt_connection_open():
     with pytest.raises(ServiceUnavailable):
-        Bolt.open(("localhost", 9999), auth=("test", "test"))
+        Bolt.open(
+            ("localhost", 9999), auth=("test", "test")
+        )
 
 
 @mark_sync_test
 def test_bolt_connection_open_timeout():
     with pytest.raises(ServiceUnavailable):
-        Bolt.open(("localhost", 9999), auth=("test", "test"),
-                             timeout=1)
+        Bolt.open(
+            ("localhost", 9999), auth=("test", "test"), deadline=Deadline(1)
+        )
 
 
 @mark_sync_test
@@ -120,7 +123,9 @@ def test_bolt_connection_ping():
 
 @mark_sync_test
 def test_bolt_connection_ping_timeout():
-    protocol_version = Bolt.ping(("localhost", 9999), timeout=1)
+    protocol_version = Bolt.ping(
+        ("localhost", 9999), deadline=Deadline(1)
+    )
     assert protocol_version is None