diff --git a/doc/source/whatsnew/v1.2.0.rst b/doc/source/whatsnew/v1.2.0.rst
index a9b4ad2e5374a..f54fa9d98a592 100644
--- a/doc/source/whatsnew/v1.2.0.rst
+++ b/doc/source/whatsnew/v1.2.0.rst
@@ -402,6 +402,7 @@ Numeric
 - Bug in :class:`DataFrame` arithmetic ops incorrectly accepting keyword arguments (:issue:`36843`)
 - Bug in :class:`IntervalArray` comparisons with :class:`Series` not returning :class:`Series` (:issue:`36908`)
 - Bug in :class:`DataFrame` allowing arithmetic operations with list of array-likes with undefined results. Behavior changed to raising ``ValueError`` (:issue:`36702`)
+- Bug in :meth:`DataFrame.std`` with ``timedelta64`` dtype and ``skipna=False`` (:issue:`37392`)
 
 Conversion
 ^^^^^^^^^^
diff --git a/pandas/core/nanops.py b/pandas/core/nanops.py
index 83399a87e5667..b101da196fdd8 100644
--- a/pandas/core/nanops.py
+++ b/pandas/core/nanops.py
@@ -228,7 +228,7 @@ def _maybe_get_mask(
             # Boolean data cannot contain nulls, so signal via mask being None
             return None
 
-        if skipna:
+        if skipna or needs_i8_conversion(values.dtype):
             mask = isna(values)
 
     return mask
diff --git a/pandas/tests/arrays/test_timedeltas.py b/pandas/tests/arrays/test_timedeltas.py
index 976f5d0c90f19..f67a554a435ef 100644
--- a/pandas/tests/arrays/test_timedeltas.py
+++ b/pandas/tests/arrays/test_timedeltas.py
@@ -3,6 +3,7 @@
 
 import pandas as pd
 import pandas._testing as tm
+from pandas.core import nanops
 from pandas.core.arrays import TimedeltaArray
 
 
@@ -288,12 +289,19 @@ def test_std(self):
         assert isinstance(result, pd.Timedelta)
         assert result == expected
 
+        result = nanops.nanstd(np.asarray(arr), skipna=True)
+        assert isinstance(result, pd.Timedelta)
+        assert result == expected
+
         result = arr.std(skipna=False)
         assert result is pd.NaT
 
         result = tdi.std(skipna=False)
         assert result is pd.NaT
 
+        result = nanops.nanstd(np.asarray(arr), skipna=False)
+        assert result is pd.NaT
+
     def test_median(self):
         tdi = pd.TimedeltaIndex(["0H", "3H", "NaT", "5H06m", "0H", "2H"])
         arr = tdi.array
@@ -307,8 +315,8 @@ def test_median(self):
         assert isinstance(result, pd.Timedelta)
         assert result == expected
 
-        result = arr.std(skipna=False)
+        result = arr.median(skipna=False)
         assert result is pd.NaT
 
-        result = tdi.std(skipna=False)
+        result = tdi.median(skipna=False)
         assert result is pd.NaT
diff --git a/pandas/tests/frame/test_analytics.py b/pandas/tests/frame/test_analytics.py
index ddca67306d804..ee4da37ce10f3 100644
--- a/pandas/tests/frame/test_analytics.py
+++ b/pandas/tests/frame/test_analytics.py
@@ -753,6 +753,22 @@ def test_operators_timedelta64(self):
         assert df["off1"].dtype == "timedelta64[ns]"
         assert df["off2"].dtype == "timedelta64[ns]"
 
+    def test_std_timedelta64_skipna_false(self):
+        # GH#37392
+        tdi = pd.timedelta_range("1 Day", periods=10)
+        df = DataFrame({"A": tdi, "B": tdi})
+        df.iloc[-2, -1] = pd.NaT
+
+        result = df.std(skipna=False)
+        expected = Series(
+            [df["A"].std(), pd.NaT], index=["A", "B"], dtype="timedelta64[ns]"
+        )
+        tm.assert_series_equal(result, expected)
+
+        result = df.std(axis=1, skipna=False)
+        expected = Series([pd.Timedelta(0)] * 8 + [pd.NaT, pd.Timedelta(0)])
+        tm.assert_series_equal(result, expected)
+
     def test_sum_corner(self):
         empty_frame = DataFrame()