diff --git a/pandas/core/frame.py b/pandas/core/frame.py
index 803d1c914c954..c5d50558b595d 100644
--- a/pandas/core/frame.py
+++ b/pandas/core/frame.py
@@ -5192,7 +5192,7 @@ def rename(
         2  2  5
         4  3  6
         """
-        return super().rename(
+        return super()._rename(
             mapper=mapper,
             index=index,
             columns=columns,
diff --git a/pandas/core/generic.py b/pandas/core/generic.py
index 4aff7acc4c6fb..b9169c8595324 100644
--- a/pandas/core/generic.py
+++ b/pandas/core/generic.py
@@ -989,7 +989,7 @@ def squeeze(self, axis=None):
     # ----------------------------------------------------------------------
     # Rename
 
-    def rename(
+    def _rename(
         self: NDFrameT,
         mapper: Renamer | None = None,
         *,
@@ -4418,7 +4418,7 @@ def add_prefix(self: NDFrameT, prefix: str) -> NDFrameT:
         # expected "NDFrameT")
         # error: Argument 1 to "rename" of "NDFrame" has incompatible type
         # "**Dict[str, partial[str]]"; expected "Union[str, int, None]"
-        return self.rename(**mapper)  # type: ignore[return-value, arg-type]
+        return self._rename(**mapper)  # type: ignore[return-value, arg-type]
 
     @final
     def add_suffix(self: NDFrameT, suffix: str) -> NDFrameT:
@@ -4482,7 +4482,7 @@ def add_suffix(self: NDFrameT, suffix: str) -> NDFrameT:
         # expected "NDFrameT")
         # error: Argument 1 to "rename" of "NDFrame" has incompatible type
         # "**Dict[str, partial[str]]"; expected "Union[str, int, None]"
-        return self.rename(**mapper)  # type: ignore[return-value, arg-type]
+        return self._rename(**mapper)  # type: ignore[return-value, arg-type]
 
     def sort_values(
         self,
diff --git a/pandas/core/series.py b/pandas/core/series.py
index ffa31b4f66211..7c04dd005fb96 100644
--- a/pandas/core/series.py
+++ b/pandas/core/series.py
@@ -4468,10 +4468,8 @@ def align(
 
     def rename(
         self,
-        mapper=None,
-        *,
         index=None,
-        columns=None,
+        *,
         axis=None,
         copy=True,
         inplace=False,
@@ -4493,7 +4491,7 @@ def rename(
         ----------
         axis : {0 or "index"}
             Unused. Accepted for compatibility with DataFrame method only.
-        mapper : scalar, hashable sequence, dict-like or function, optional
+        index : scalar, hashable sequence, dict-like or function, optional
             Functions or dict-like are transformations to apply to
             the index.
             Scalar or hashable sequence-like will alter the ``Series.name``
@@ -4541,16 +4539,12 @@ def rename(
             # Make sure we raise if an invalid 'axis' is passed.
             axis = self._get_axis_number(axis)
 
-        if index is not None and mapper is not None:
-            raise TypeError("Cannot specify both 'mapper' and 'index'")
-        if mapper is None:
-            mapper = index
-        if callable(mapper) or is_dict_like(mapper):
-            return super().rename(
-                mapper, copy=copy, inplace=inplace, level=level, errors=errors
+        if callable(index) or is_dict_like(index):
+            return super()._rename(
+                index, copy=copy, inplace=inplace, level=level, errors=errors
             )
         else:
-            return self._set_name(mapper, inplace=inplace)
+            return self._set_name(index, inplace=inplace)
 
     @overload
     def set_axis(
diff --git a/pandas/tests/series/methods/test_rename.py b/pandas/tests/series/methods/test_rename.py
index 3425dd8f019e7..90c8f775586e6 100644
--- a/pandas/tests/series/methods/test_rename.py
+++ b/pandas/tests/series/methods/test_rename.py
@@ -105,12 +105,6 @@ def test_rename_callable(self):
 
         assert result.name == expected.name
 
-    def test_rename_method_and_index(self):
-        # GH 40977
-        ser = Series([1, 2])
-        with pytest.raises(TypeError, match="Cannot specify both 'mapper' and 'index'"):
-            ser.rename(str, index=str)
-
     def test_rename_none(self):
         # GH 40977
         ser = Series([1, 2], name="foo")