diff --git a/doc/source/whatsnew/v3.0.0.rst b/doc/source/whatsnew/v3.0.0.rst
index 184ca581902ee..8873f7c1a8fe8 100644
--- a/doc/source/whatsnew/v3.0.0.rst
+++ b/doc/source/whatsnew/v3.0.0.rst
@@ -421,6 +421,7 @@ Other Deprecations
 - Deprecated lowercase strings ``w``, ``w-mon``, ``w-tue``, etc. denoting frequencies in :class:`Week` in favour of ``W``, ``W-MON``, ``W-TUE``, etc. (:issue:`58998`)
 - Deprecated parameter ``method`` in :meth:`DataFrame.reindex_like` / :meth:`Series.reindex_like` (:issue:`58667`)
 - Deprecated strings ``w``, ``d``, ``MIN``, ``MS``, ``US`` and ``NS`` denoting units in :class:`Timedelta` in favour of ``W``, ``D``, ``min``, ``ms``, ``us`` and ``ns`` (:issue:`59051`)
+- Deprecated the ``arg`` parameter of ``Series.map``; pass the added ``func`` argument instead. (:issue:`61260`)
 - Deprecated using ``epoch`` date format in :meth:`DataFrame.to_json` and :meth:`Series.to_json`, use ``iso`` instead. (:issue:`57063`)
 
 .. ---------------------------------------------------------------------------
diff --git a/pandas/core/series.py b/pandas/core/series.py
index 03a2ce85a08c9..d6a982c65e9fd 100644
--- a/pandas/core/series.py
+++ b/pandas/core/series.py
@@ -52,6 +52,9 @@
     doc,
     set_module,
 )
+from pandas.util._exceptions import (
+    find_stack_level,
+)
 from pandas.util._validators import (
     validate_ascending,
     validate_bool_kwarg,
@@ -4320,7 +4323,7 @@ def unstack(
 
     def map(
         self,
-        arg: Callable | Mapping | Series,
+        func: Callable | Mapping | Series | None = None,
         na_action: Literal["ignore"] | None = None,
         **kwargs,
     ) -> Series:
@@ -4333,8 +4336,8 @@ def map(
 
         Parameters
         ----------
-        arg : function, collections.abc.Mapping subclass or Series
-            Mapping correspondence.
+        func : function, collections.abc.Mapping subclass or Series
+            Function or mapping correspondence.
         na_action : {None, 'ignore'}, default None
             If 'ignore', propagate NaN values, without passing them to the
             mapping correspondence.
@@ -4404,9 +4407,22 @@ def map(
         3  I am a rabbit
         dtype: object
         """
-        if callable(arg):
-            arg = functools.partial(arg, **kwargs)
-        new_values = self._map_values(arg, na_action=na_action)
+        if func is None:
+            if "arg" in kwargs:
+                # `.map(arg=my_func)`
+                func = kwargs.pop("arg")
+                warnings.warn(
+                    "The parameter `arg` has been renamed to `func`, and it "
+                    "will stop being supported in a future version of pandas.",
+                    FutureWarning,
+                    stacklevel=find_stack_level(),
+                )
+            else:
+                raise ValueError("The `func` parameter is required")
+
+        if callable(func):
+            func = functools.partial(func, **kwargs)
+        new_values = self._map_values(func, na_action=na_action)
         return self._constructor(new_values, index=self.index, copy=False).__finalize__(
             self, method="map"
         )
diff --git a/pandas/tests/series/methods/test_map.py b/pandas/tests/series/methods/test_map.py
index 84b60a2afe6eb..384b7ce3dc985 100644
--- a/pandas/tests/series/methods/test_map.py
+++ b/pandas/tests/series/methods/test_map.py
@@ -604,3 +604,27 @@ def test_map_kwargs():
     result = Series([2, 4, 5]).map(lambda x, y: x + y, y=2)
     expected = Series([4, 6, 7])
     tm.assert_series_equal(result, expected)
+
+
+def test_map_arg_as_kwarg():
+    with tm.assert_produces_warning(
+        FutureWarning, match="`arg` has been renamed to `func`"
+    ):
+        Series([1, 2]).map(arg={})
+
+
+def test_map_func_and_arg():
+    # `arg`is considered a normal kwarg that should be passed to the function
+    result = Series([1, 2]).map(lambda _, arg: arg, arg=3)
+    expected = Series([3, 3])
+    tm.assert_series_equal(result, expected)
+
+
+def test_map_no_func_or_arg():
+    with pytest.raises(ValueError, match="The `func` parameter is required"):
+        Series([1, 2]).map()
+
+
+def test_map_func_is_none():
+    with pytest.raises(ValueError, match="The `func` parameter is required"):
+        Series([1, 2]).map(func=None)