diff --git a/Lib/_abcoll.py b/Lib/_abcoll.py
index cac06e0825f..0957553613b 100644
--- a/Lib/_abcoll.py
+++ b/Lib/_abcoll.py
@@ -90,7 +90,8 @@ class Iterator(Iterable):
     @classmethod
     def __subclasshook__(cls, C):
         if cls is Iterator:
-            if any("__next__" in B.__dict__ for B in C.__mro__):
+            if (any("__next__" in B.__dict__ for B in C.__mro__) and
+                any("__iter__" in B.__dict__ for B in C.__mro__)):
                 return True
         return NotImplemented
 
diff --git a/Lib/test/test_collections.py b/Lib/test/test_collections.py
index dc61fa41470..988db92e16c 100644
--- a/Lib/test/test_collections.py
+++ b/Lib/test/test_collections.py
@@ -328,7 +328,14 @@ class TestOneTrickPonyABCs(ABCTestCase):
         for x in samples:
             self.assertTrue(isinstance(x, Iterator), repr(x))
             self.assertTrue(issubclass(type(x), Iterator), repr(type(x)))
-        self.validate_abstract_methods(Iterator, '__next__')
+        self.validate_abstract_methods(Iterator, '__next__', '__iter__')
+
+        # Issue 10565
+        class NextOnly:
+            def __next__(self):
+                yield 1
+                raise StopIteration
+        self.assertFalse(isinstance(NextOnly(), Iterator))
 
     def test_Sized(self):
         non_samples = [None, 42, 3.14, 1j,
diff --git a/Misc/NEWS b/Misc/NEWS
index 1167fff329b..11a26d4acb1 100644
--- a/Misc/NEWS
+++ b/Misc/NEWS
@@ -28,6 +28,9 @@ Core and Builtins
 Library
 -------
 
+- Issue #10565: The collections.Iterator ABC now checks for both
+  __iter__ and __next__.
+
 - Issue #10561: In pdb, clear the breakpoints by the breakpoint number.
 
 - Issue #10459: Update CJK character names to Unicode 5.1.