diff --git a/pandas/core/dtypes/cast.py b/pandas/core/dtypes/cast.py
index 929dfa5c12078..5f1c9c1889240 100644
--- a/pandas/core/dtypes/cast.py
+++ b/pandas/core/dtypes/cast.py
@@ -1811,8 +1811,17 @@ def maybe_cast_to_integer_array(
     # doesn't handle `uint64` correctly.
     arr = np.asarray(arr)
 
-    if is_unsigned_integer_dtype(dtype) and (arr < 0).any():
-        raise OverflowError("Trying to coerce negative values to unsigned integers")
+    if is_unsigned_integer_dtype(dtype):
+        try:
+            if (arr < 0).any():
+                raise OverflowError(
+                    "Trying to coerce negative values to unsigned integers"
+                )
+        except TypeError:
+            if (casted < 0).any():
+                raise OverflowError(
+                    "Trying to coerce negative values to unsigned integers"
+                )
 
     if is_float_dtype(arr.dtype):
         if not np.isfinite(arr).all():
@@ -1823,7 +1832,7 @@ def maybe_cast_to_integer_array(
     if is_object_dtype(arr.dtype):
         raise ValueError("Trying to coerce float values to integers")
 
-    if casted.dtype < arr.dtype:
+    if casted.dtype < arr.dtype or is_string_dtype(arr.dtype):
         # GH#41734 e.g. [1, 200, 923442] and dtype="int8" -> overflows
         warnings.warn(
             f"Values are too large to be losslessly cast to {dtype}. "
diff --git a/pandas/tests/series/test_constructors.py b/pandas/tests/series/test_constructors.py
index 3b7ae28be68fa..b33e3d88b7748 100644
--- a/pandas/tests/series/test_constructors.py
+++ b/pandas/tests/series/test_constructors.py
@@ -1895,6 +1895,20 @@ def test_constructor_bool_dtype_missing_values(self):
         expected = Series(True, index=[0], dtype="bool")
         tm.assert_series_equal(result, expected)
 
+    def test_constructor_int64_dtype(self, any_int_dtype):
+        # GH-44923
+        result = Series(["0", "1", "2"], dtype=any_int_dtype)
+        expected = Series([0, 1, 2], dtype=any_int_dtype)
+        tm.assert_series_equal(result, expected)
+
+    def test_constructor_float64_dtype(self, any_float_dtype):
+        # GH-44923
+        if any_float_dtype in ["Float32", "Float64"]:
+            pytest.xfail(reason="Cannot be casted to FloatDtype Series")
+        result = Series(["-1", "0", "1", "2"], dtype=any_float_dtype)
+        expected = Series([-1.0, 0.0, 1.0, 2.0], dtype=any_float_dtype)
+        tm.assert_series_equal(result, expected)
+
     @pytest.mark.filterwarnings(
         "ignore:elementwise comparison failed:DeprecationWarning"
     )