diff --git a/doc/source/whatsnew/v1.3.0.rst b/doc/source/whatsnew/v1.3.0.rst
index fe88cf93886ce..66d9e720153bd 100644
--- a/doc/source/whatsnew/v1.3.0.rst
+++ b/doc/source/whatsnew/v1.3.0.rst
@@ -679,6 +679,7 @@ Deprecations
 - Deprecated the ``convert_float`` optional argument in :func:`read_excel` and :meth:`ExcelFile.parse` (:issue:`41127`)
 - Deprecated behavior of :meth:`DatetimeIndex.union` with mixed timezones; in a future version both will be cast to UTC instead of object dtype (:issue:`39328`)
 - Deprecated using ``usecols`` with out of bounds indices for ``read_csv`` with ``engine="c"`` (:issue:`25623`)
+- Deprecated passing arguments as positional in :meth:`Index.set_names` and :meth:`MultiIndex.set_names` (except for ``names``) (:issue:`41485`)
 - Deprecated passing arguments as positional in :meth:`DataFrame.clip` and :meth:`Series.clip` (other than ``"upper"`` and ``"lower"``) (:issue:`41485`)
 - Deprecated special treatment of lists with first element a Categorical in the :class:`DataFrame` constructor; pass as ``pd.DataFrame({col: categorical, ...})`` instead (:issue:`38845`)
 - Deprecated passing arguments as positional (except for ``"method"``) in :meth:`DataFrame.interpolate` and :meth:`Series.interpolate` (:issue:`41485`)
diff --git a/pandas/core/indexes/base.py b/pandas/core/indexes/base.py
index 2a6f044288fea..2a50ebd959ace 100644
--- a/pandas/core/indexes/base.py
+++ b/pandas/core/indexes/base.py
@@ -1538,7 +1538,7 @@ def _set_names(self, values, level=None) -> None:
 
     names = property(fset=_set_names, fget=_get_names)
 
-    @final
+    @deprecate_nonkeyword_arguments(version=None, allowed_args=["self", "names"])
     def set_names(self, names, level=None, inplace: bool = False):
         """
         Set Index or MultiIndex name.
diff --git a/pandas/core/indexes/multi.py b/pandas/core/indexes/multi.py
index 6a13ffa5a376b..1362679ae0064 100644
--- a/pandas/core/indexes/multi.py
+++ b/pandas/core/indexes/multi.py
@@ -292,7 +292,6 @@ class MultiIndex(Index):
     _levels = FrozenList()
     _codes = FrozenList()
     _comparables = ["names"]
-    rename = Index.set_names
 
     sortorder: int | None
 
@@ -3585,7 +3584,9 @@ def _get_reconciled_name_object(self, other) -> MultiIndex:
         """
         names = self._maybe_match_names(other)
         if self.names != names:
-            return self.rename(names)
+            # Incompatible return value type (got "Optional[MultiIndex]", expected
+            # "MultiIndex")
+            return self.rename(names)  # type: ignore[return-value]
         return self
 
     def _maybe_match_names(self, other):
@@ -3784,6 +3785,12 @@ def isin(self, values, level=None) -> np.ndarray:
                 return np.zeros(len(levs), dtype=np.bool_)
             return levs.isin(values)
 
+    @deprecate_nonkeyword_arguments(version=None, allowed_args=["self", "names"])
+    def set_names(self, names, level=None, inplace: bool = False) -> MultiIndex | None:
+        return super().set_names(names=names, level=level, inplace=inplace)
+
+    rename = set_names
+
     @deprecate_nonkeyword_arguments(version=None, allowed_args=["self"])
     def drop_duplicates(self, keep: str | bool = "first") -> MultiIndex:
         return super().drop_duplicates(keep=keep)
diff --git a/pandas/tests/indexes/multi/test_get_set.py b/pandas/tests/indexes/multi/test_get_set.py
index 9657e74c363b9..e756f95bb2bc5 100644
--- a/pandas/tests/indexes/multi/test_get_set.py
+++ b/pandas/tests/indexes/multi/test_get_set.py
@@ -345,6 +345,23 @@ def test_set_names_with_nlevel_1(inplace):
     tm.assert_index_equal(result, expected)
 
 
+def test_multi_set_names_pos_args_deprecation():
+    # GH#41485
+    idx = MultiIndex.from_product([["python", "cobra"], [2018, 2019]])
+    msg = (
+        "In a future version of pandas all arguments of MultiIndex.set_names "
+        "except for the argument 'names' will be keyword-only"
+    )
+    with tm.assert_produces_warning(FutureWarning, match=msg):
+        result = idx.set_names(["kind", "year"], None)
+    expected = MultiIndex(
+        levels=[["python", "cobra"], [2018, 2019]],
+        codes=[[0, 0, 1, 1], [0, 1, 0, 1]],
+        names=["kind", "year"],
+    )
+    tm.assert_index_equal(result, expected)
+
+
 @pytest.mark.parametrize("ordered", [True, False])
 def test_set_levels_categorical(ordered):
     # GH13854
diff --git a/pandas/tests/indexes/test_base.py b/pandas/tests/indexes/test_base.py
index f41c79bd09f67..f75e4af888643 100644
--- a/pandas/tests/indexes/test_base.py
+++ b/pandas/tests/indexes/test_base.py
@@ -1740,6 +1740,19 @@ def test_construct_from_memoryview(klass, extra_kwargs):
     tm.assert_index_equal(result, expected)
 
 
+def test_index_set_names_pos_args_deprecation():
+    # GH#41485
+    idx = Index([1, 2, 3, 4])
+    msg = (
+        "In a future version of pandas all arguments of Index.set_names "
+        "except for the argument 'names' will be keyword-only"
+    )
+    with tm.assert_produces_warning(FutureWarning, match=msg):
+        result = idx.set_names("quarter", None)
+    expected = Index([1, 2, 3, 4], name="quarter")
+    tm.assert_index_equal(result, expected)
+
+
 def test_drop_duplicates_pos_args_deprecation():
     # GH#41485
     idx = Index([1, 2, 3, 1])