diff --git a/pandas/core/generic.py b/pandas/core/generic.py
index c893e9ce3d9a9..e86427b8cd29d 100644
--- a/pandas/core/generic.py
+++ b/pandas/core/generic.py
@@ -1209,7 +1209,8 @@ class  name
         axes, kwargs = self._construct_axes_from_arguments(
             (), kwargs, sentinel=lib.no_default
         )
-        copy = kwargs.pop("copy", True)
+        copy: bool_t | None = kwargs.pop("copy", None)
+
         inplace = kwargs.pop("inplace", False)
         axis = kwargs.pop("axis", 0)
         if axis is not None:
@@ -1229,7 +1230,9 @@ class  name
                 is_list_like(mapper) and not is_dict_like(mapper)
             )
             if non_mapper:
-                return self._set_axis_name(mapper, axis=axis, inplace=inplace)
+                return self._set_axis_name(
+                    mapper, axis=axis, inplace=inplace, copy=copy
+                )
             else:
                 raise ValueError("Use `.rename` to alter labels with a mapper.")
         else:
@@ -1248,13 +1251,15 @@ class  name
                     f = common.get_rename_function(v)
                     curnames = self._get_axis(axis).names
                     newnames = [f(name) for name in curnames]
-                result._set_axis_name(newnames, axis=axis, inplace=True)
+                result._set_axis_name(newnames, axis=axis, inplace=True, copy=copy)
             if not inplace:
                 return result
             return None
 
     @final
-    def _set_axis_name(self, name, axis: Axis = 0, inplace: bool_t = False):
+    def _set_axis_name(
+        self, name, axis: Axis = 0, inplace: bool_t = False, copy: bool_t | None = True
+    ):
         """
         Set the name(s) of the axis.
 
@@ -1267,6 +1272,8 @@ def _set_axis_name(self, name, axis: Axis = 0, inplace: bool_t = False):
             and the value 1 or 'columns' specifies columns.
         inplace : bool, default False
             If `True`, do operation inplace and return None.
+        copy:
+            Whether to make a copy of the result.
 
         Returns
         -------
@@ -1308,7 +1315,7 @@ def _set_axis_name(self, name, axis: Axis = 0, inplace: bool_t = False):
         idx = self._get_axis(axis).set_names(name)
 
         inplace = validate_bool_kwarg(inplace, "inplace")
-        renamed = self if inplace else self.copy()
+        renamed = self if inplace else self.copy(deep=copy)
         if axis == 0:
             renamed.index = idx
         else:
diff --git a/pandas/tests/copy_view/test_methods.py b/pandas/tests/copy_view/test_methods.py
index 878f1d8089d33..6e49ac99c1dd1 100644
--- a/pandas/tests/copy_view/test_methods.py
+++ b/pandas/tests/copy_view/test_methods.py
@@ -3,6 +3,7 @@
 
 from pandas import (
     DataFrame,
+    Index,
     MultiIndex,
     Series,
 )
@@ -456,3 +457,21 @@ def test_series_set_axis(using_copy_on_write):
     ser2.iloc[0] = 0
     assert not np.shares_memory(ser2, ser)
     tm.assert_series_equal(ser, ser_orig)
+
+
+@pytest.mark.parametrize("copy_kwargs", [{"copy": True}, {}])
+@pytest.mark.parametrize("kwargs", [{"mapper": "test"}, {"index": "test"}])
+def test_rename_axis(using_copy_on_write, kwargs, copy_kwargs):
+    df = DataFrame({"a": [1, 2, 3, 4]}, index=Index([1, 2, 3, 4], name="a"))
+    df_orig = df.copy()
+    df2 = df.rename_axis(**kwargs, **copy_kwargs)
+
+    if using_copy_on_write and not copy_kwargs:
+        assert np.shares_memory(get_array(df2, "a"), get_array(df, "a"))
+    else:
+        assert not np.shares_memory(get_array(df2, "a"), get_array(df, "a"))
+
+    df2.iloc[0, 0] = 0
+    if using_copy_on_write:
+        assert not np.shares_memory(get_array(df2, "a"), get_array(df, "a"))
+    tm.assert_frame_equal(df, df_orig)