diff --git a/pandas/core/ops/dispatch.py b/pandas/core/ops/dispatch.py
index 637f0fa1d52e9..a7dcdd4f9d585 100644
--- a/pandas/core/ops/dispatch.py
+++ b/pandas/core/ops/dispatch.py
@@ -65,8 +65,12 @@ def should_series_dispatch(left, right, op):
     ldtype = left.dtypes.iloc[0]
     rdtype = right.dtypes.iloc[0]
 
-    if (is_timedelta64_dtype(ldtype) and is_integer_dtype(rdtype)) or (
-        is_timedelta64_dtype(rdtype) and is_integer_dtype(ldtype)
+    if (
+        is_timedelta64_dtype(ldtype)
+        and (is_integer_dtype(rdtype) or is_object_dtype(rdtype))
+    ) or (
+        is_timedelta64_dtype(rdtype)
+        and (is_integer_dtype(ldtype) or is_object_dtype(ldtype))
     ):
         # numpy integer dtypes as timedelta64 dtypes in this scenario
         return True
diff --git a/pandas/tests/arithmetic/test_timedelta64.py b/pandas/tests/arithmetic/test_timedelta64.py
index b2ea6d8b92fdd..e3eec1f781948 100644
--- a/pandas/tests/arithmetic/test_timedelta64.py
+++ b/pandas/tests/arithmetic/test_timedelta64.py
@@ -541,9 +541,10 @@ def test_tda_add_sub_index(self):
         expected = tdi - tdi
         tm.assert_index_equal(result, expected)
 
-    def test_tda_add_dt64_object_array(self, box_df_fail, tz_naive_fixture):
+    def test_tda_add_dt64_object_array(self, box_with_array, tz_naive_fixture):
         # Result should be cast back to DatetimeArray
-        box = box_df_fail
+        box = box_with_array
+
         dti = pd.date_range("2016-01-01", periods=3, tz=tz_naive_fixture)
         dti = dti._with_freq(None)
         tdi = dti - dti
@@ -1396,33 +1397,40 @@ def test_td64arr_sub_offset_array(self, box_with_array):
             res = tdi - other
         tm.assert_equal(res, expected)
 
-    def test_td64arr_with_offset_series(self, names, box_df_fail):
+    def test_td64arr_with_offset_series(self, names, box_with_array):
         # GH#18849
-        box = box_df_fail
+        box = box_with_array
         box2 = Series if box in [pd.Index, tm.to_array] else box
-        exname = names[2] if box is not tm.to_array else names[1]
+
+        if box is pd.DataFrame:
+            # Since we are operating with a DataFrame and a non-DataFrame,
+            # the non-DataFrame is cast to Series and its name ignored.
+            exname = names[0]
+        elif box is tm.to_array:
+            exname = names[1]
+        else:
+            exname = names[2]
 
         tdi = TimedeltaIndex(["1 days 00:00:00", "3 days 04:00:00"], name=names[0])
         other = Series([pd.offsets.Hour(n=1), pd.offsets.Minute(n=-2)], name=names[1])
 
         expected_add = Series([tdi[n] + other[n] for n in range(len(tdi))], name=exname)
-        tdi = tm.box_expected(tdi, box)
+        obj = tm.box_expected(tdi, box)
         expected_add = tm.box_expected(expected_add, box2)
 
         with tm.assert_produces_warning(PerformanceWarning):
-            res = tdi + other
+            res = obj + other
         tm.assert_equal(res, expected_add)
 
         with tm.assert_produces_warning(PerformanceWarning):
-            res2 = other + tdi
+            res2 = other + obj
         tm.assert_equal(res2, expected_add)
 
-        # TODO: separate/parametrize add/sub test?
         expected_sub = Series([tdi[n] - other[n] for n in range(len(tdi))], name=exname)
         expected_sub = tm.box_expected(expected_sub, box2)
 
         with tm.assert_produces_warning(PerformanceWarning):
-            res3 = tdi - other
+            res3 = obj - other
         tm.assert_equal(res3, expected_sub)
 
     @pytest.mark.parametrize("obox", [np.array, pd.Index, pd.Series])