diff --git a/doc/source/whatsnew/v1.2.0.rst b/doc/source/whatsnew/v1.2.0.rst
index ccaae9f996425..2afa1f1a6199e 100644
--- a/doc/source/whatsnew/v1.2.0.rst
+++ b/doc/source/whatsnew/v1.2.0.rst
@@ -228,6 +228,7 @@ Datetimelike
 - Bug in :class:`DateOffset` where attributes reconstructed from pickle files differ from original objects when input values exceed normal ranges (e.g months=12) (:issue:`34511`)
 - Bug in :meth:`DatetimeIndex.get_slice_bound` where ``datetime.date`` objects were not accepted or naive :class:`Timestamp` with a tz-aware :class:`DatetimeIndex` (:issue:`35690`)
 - Bug in :meth:`DatetimeIndex.slice_locs` where ``datetime.date`` objects were not accepted (:issue:`34077`)
+- Bug in :meth:`DatetimeIndex.searchsorted`, :meth:`TimedeltaIndex.searchsorted`, and :meth:`Series.searchsorted` with ``datetime64`` or ``timedelta64`` dtype placement of ``NaT`` values being inconsistent with ``NumPy`` (:issue:`36176`)
 
 Timedelta
 ^^^^^^^^^
diff --git a/pandas/core/arrays/datetimelike.py b/pandas/core/arrays/datetimelike.py
index a218745db0a44..e81904713ea5b 100644
--- a/pandas/core/arrays/datetimelike.py
+++ b/pandas/core/arrays/datetimelike.py
@@ -858,7 +858,8 @@ def _validate_searchsorted_value(self, value):
             # TODO: cast_str?  we accept it for scalar
             value = self._validate_listlike(value, "searchsorted")
 
-        return self._unbox(value)
+        rv = self._unbox(value)
+        return self._rebox_native(rv)
 
     def _validate_setitem_value(self, value):
         msg = (
@@ -937,9 +938,7 @@ def searchsorted(self, value, side="left", sorter=None):
             Array of insertion points with the same shape as `value`.
         """
         value = self._validate_searchsorted_value(value)
-
-        # TODO: Use datetime64 semantics for sorting, xref GH#29844
-        return self.asi8.searchsorted(value, side=side, sorter=sorter)
+        return self._data.searchsorted(value, side=side, sorter=sorter)
 
     def value_counts(self, dropna=False):
         """
diff --git a/pandas/tests/arrays/test_datetimelike.py b/pandas/tests/arrays/test_datetimelike.py
index b1ab700427c28..292557fc04258 100644
--- a/pandas/tests/arrays/test_datetimelike.py
+++ b/pandas/tests/arrays/test_datetimelike.py
@@ -241,10 +241,15 @@ def test_searchsorted(self):
         expected = np.array([2, 3], dtype=np.intp)
         tm.assert_numpy_array_equal(result, expected)
 
-        # Following numpy convention, NaT goes at the beginning
-        #  (unlike NaN which goes at the end)
+        # GH#29884 match numpy convention on whether NaT goes
+        #  at the end or the beginning
         result = arr.searchsorted(pd.NaT)
-        assert result == 0
+        if _np_version_under1p18 or self.array_cls is PeriodArray:
+            # Following numpy convention, NaT goes at the beginning
+            #  (unlike NaN which goes at the end)
+            assert result == 0
+        else:
+            assert result == 10
 
     def test_getitem_2d(self, arr1d):
         # 2d slicing on a 1D array