diff --git a/doc/source/whatsnew/v0.23.1.txt b/doc/source/whatsnew/v0.23.1.txt
index 9c29c34adb7dd..974527624a312 100644
--- a/doc/source/whatsnew/v0.23.1.txt
+++ b/doc/source/whatsnew/v0.23.1.txt
@@ -86,8 +86,10 @@ Indexing
 - Bug in :meth:`Series.reset_index` where appropriate error was not raised with an invalid level name (:issue:`20925`)
 - Bug in :func:`interval_range` when ``start``/``periods`` or ``end``/``periods`` are specified with float ``start`` or ``end`` (:issue:`21161`)
 - Bug in :meth:`MultiIndex.set_names` where error raised for a ``MultiIndex`` with ``nlevels == 1`` (:issue:`21149`)
+- Bug in :attr:`DatetimeIndex.date` where an incorrect date is returned when the input date has a non-UTC timezone (:issue:`21230`)
 - Bug in :class:`IntervalIndex` constructors where creating an ``IntervalIndex`` from categorical data was not fully supported (:issue:`21243`, issue:`21253`)
 - Bug in :meth:`MultiIndex.sort_index` which was not guaranteed to sort correctly with ``level=1``; this was also causing data misalignment in particular :meth:`DataFrame.stack` operations (:issue:`20994`, :issue:`20945`, :issue:`21052`)
+- Bug in :attr:`DatetimeIndex.time` where given a tz-aware Timestamp, a tz-aware Time is returned instead of tz-naive  (:issue:`21267`)
 -
 
 I/O
diff --git a/pandas/_libs/tslib.pyx b/pandas/_libs/tslib.pyx
index 17453d8af1297..0f58cfa761f21 100644
--- a/pandas/_libs/tslib.pyx
+++ b/pandas/_libs/tslib.pyx
@@ -77,7 +77,7 @@ cdef inline object create_time_from_ts(
         int64_t value, pandas_datetimestruct dts,
         object tz, object freq):
     """ convenience routine to construct a datetime.time from its parts """
-    return time(dts.hour, dts.min, dts.sec, dts.us, tz)
+    return time(dts.hour, dts.min, dts.sec, dts.us)
 
 
 def ints_to_pydatetime(ndarray[int64_t] arr, tz=None, freq=None,
diff --git a/pandas/core/indexes/datetimes.py b/pandas/core/indexes/datetimes.py
index 83950f1d71633..0ddf33cdcae73 100644
--- a/pandas/core/indexes/datetimes.py
+++ b/pandas/core/indexes/datetimes.py
@@ -2032,7 +2032,16 @@ def time(self):
         """
         Returns numpy array of datetime.time. The time part of the Timestamps.
         """
-        return libts.ints_to_pydatetime(self.asi8, self.tz, box="time")
+
+        # If the Timestamps have a timezone that is not UTC,
+        # convert them into their i8 representation while
+        # keeping their timezone and not using UTC
+        if (self.tz is not None and self.tz is not utc):
+            timestamps = self._local_timestamps()
+        else:
+            timestamps = self.asi8
+
+        return libts.ints_to_pydatetime(timestamps, box="time")
 
     @property
     def date(self):
@@ -2040,7 +2049,16 @@ def date(self):
         Returns numpy array of python datetime.date objects (namely, the date
         part of Timestamps without timezone information).
         """
-        return libts.ints_to_pydatetime(self.normalize().asi8, box="date")
+
+        # If the Timestamps have a timezone that is not UTC,
+        # convert them into their i8 representation while
+        # keeping their timezone and not using UTC
+        if (self.tz is not None and self.tz is not utc):
+            timestamps = self._local_timestamps()
+        else:
+            timestamps = self.asi8
+
+        return libts.ints_to_pydatetime(timestamps, box="date")
 
     def normalize(self):
         """
diff --git a/pandas/tests/indexes/datetimes/test_timezones.py b/pandas/tests/indexes/datetimes/test_timezones.py
index 09210d8b64d1b..573940edaa08f 100644
--- a/pandas/tests/indexes/datetimes/test_timezones.py
+++ b/pandas/tests/indexes/datetimes/test_timezones.py
@@ -2,7 +2,7 @@
 """
 Tests for DatetimeIndex timezone-related methods
 """
-from datetime import datetime, timedelta, tzinfo
+from datetime import datetime, timedelta, tzinfo, date, time
 from distutils.version import LooseVersion
 
 import pytest
@@ -706,6 +706,32 @@ def test_join_utc_convert(self, join_type):
         assert isinstance(result, DatetimeIndex)
         assert result.tz.zone == 'UTC'
 
+    @pytest.mark.parametrize("dtype", [
+        None, 'datetime64[ns, CET]',
+        'datetime64[ns, EST]', 'datetime64[ns, UTC]'
+    ])
+    def test_date_accessor(self, dtype):
+        # Regression test for GH#21230
+        expected = np.array([date(2018, 6, 4), pd.NaT])
+
+        index = DatetimeIndex(['2018-06-04 10:00:00', pd.NaT], dtype=dtype)
+        result = index.date
+
+        tm.assert_numpy_array_equal(result, expected)
+
+    @pytest.mark.parametrize("dtype", [
+        None, 'datetime64[ns, CET]',
+        'datetime64[ns, EST]', 'datetime64[ns, UTC]'
+    ])
+    def test_time_accessor(self, dtype):
+        # Regression test for GH#21267
+        expected = np.array([time(10, 20, 30), pd.NaT])
+
+        index = DatetimeIndex(['2018-06-04 10:20:30', pd.NaT], dtype=dtype)
+        result = index.time
+
+        tm.assert_numpy_array_equal(result, expected)
+
     def test_dti_drop_dont_lose_tz(self):
         # GH#2621
         ind = date_range("2012-12-01", periods=10, tz="utc")