diff --git a/pandas/core/nanops.py b/pandas/core/nanops.py
index f07f054c2216b..83067ffd64c79 100644
--- a/pandas/core/nanops.py
+++ b/pandas/core/nanops.py
@@ -718,7 +718,8 @@ def reduction(values, axis=None, skipna=True, mask=None):
                 result = np.nan
         else:
             result = getattr(values, meth)(axis)
-            if is_integer(result) and result == _int64_max:
+            if (is_integer(result) and is_datetime_or_timedelta_dtype(dtype)
+                    and result == _int64_max):
                 result = tslibs.iNaT
 
         result = _wrap_results(result, dtype)
diff --git a/pandas/tests/series/test_datetime_values.py b/pandas/tests/series/test_datetime_values.py
index cd6a81230492c..1fb7de3730b55 100644
--- a/pandas/tests/series/test_datetime_values.py
+++ b/pandas/tests/series/test_datetime_values.py
@@ -510,7 +510,18 @@ def test_dt_timetz_accessor(self, tz_naive_fixture):
         result = s.dt.timetz
         tm.assert_series_equal(result, expected)
 
-    def test_minmax_nat(self):
-        series = pd.Series([pd.NaT, pd.NaT])
-        assert series.min() is pd.NaT
-        assert series.max() is pd.NaT
+    @pytest.mark.parametrize('nat', [
+        pd.Series([pd.NaT, pd.NaT]),
+        pd.Series([pd.NaT, pd.Timedelta('nat')]),
+        pd.Series([pd.Timedelta('nat'), pd.Timedelta('nat')])])
+    def test_minmax_nat_series(self, nat):
+        assert nat.min() is pd.NaT
+        assert nat.max() is pd.NaT
+
+    @pytest.mark.parametrize('nat', [
+        pd.DataFrame([pd.NaT, pd.NaT]),
+        pd.DataFrame([pd.NaT, pd.Timedelta('nat')]),
+        pd.DataFrame([pd.Timedelta('nat'), pd.Timedelta('nat')])])
+    def test_minmax_nat_dataframe(self, nat):
+        assert nat.min()[0] is pd.NaT
+        assert nat.max()[0] is pd.NaT