diff --git a/doc/source/whatsnew/v0.22.0.txt b/doc/source/whatsnew/v0.22.0.txt
index 3d8e592c49221..e85ba505887b4 100644
--- a/doc/source/whatsnew/v0.22.0.txt
+++ b/doc/source/whatsnew/v0.22.0.txt
@@ -40,6 +40,7 @@ Backwards incompatible API changes
 Other API Changes
 ^^^^^^^^^^^^^^^^^
 
+- ``NaT`` division with :class:`datetime.timedelta` will now return ``NaN`` instead of raising (:issue:`17876`)
 - :class:`Timestamp` will no longer silently ignore unused or invalid `tz` or `tzinfo` arguments (:issue:`17690`)
 -
 -
diff --git a/pandas/_libs/tslib.pyx b/pandas/_libs/tslib.pyx
index 0f39c2947dfa0..53cde4f9b6b65 100644
--- a/pandas/_libs/tslib.pyx
+++ b/pandas/_libs/tslib.pyx
@@ -1448,7 +1448,7 @@ _nat_scalar_rules[Py_GE] = False
 
 
 cdef _nat_divide_op(self, other):
-    if (isinstance(other, Timedelta) or
+    if (PyDelta_Check(other) or
         is_timedelta64_object(other) or other is NaT):
         return np.nan
     if is_integer_object(other) or is_float_object(other):
@@ -1456,7 +1456,7 @@ cdef _nat_divide_op(self, other):
     return NotImplemented
 
 cdef _nat_rdivide_op(self, other):
-    if isinstance(other, Timedelta):
+    if PyDelta_Check(other):
         return np.nan
     return NotImplemented
 
diff --git a/pandas/tests/scalar/test_nat.py b/pandas/tests/scalar/test_nat.py
index 0e69371511294..7194849f19ebb 100644
--- a/pandas/tests/scalar/test_nat.py
+++ b/pandas/tests/scalar/test_nat.py
@@ -248,6 +248,8 @@ def test_nat_arithmetic():
         assert left + right is NaT
         assert right - left is NaT
         assert left - right is NaT
+        assert np.isnan(left / right)
+        assert np.isnan(right / left)
 
     # GH 11718
     t_utc = Timestamp('2014-01-01', tz='UTC')