diff --git a/Lib/test/test_heapq.py b/Lib/test/test_heapq.py
index 861ba7540df..6902573e8fa 100644
--- a/Lib/test/test_heapq.py
+++ b/Lib/test/test_heapq.py
@@ -432,6 +432,37 @@ class TestErrorHandling:
         with self.assertRaises((IndexError, RuntimeError)):
             self.module.heappop(heap)
 
+    def test_comparison_operator_modifiying_heap(self):
+        # See bpo-39421: Strong references need to be taken
+        # when comparing objects as they can alter the heap
+        class EvilClass(int):
+            def __lt__(self, o):
+                heap.clear()
+                return NotImplemented
+
+        heap = []
+        self.module.heappush(heap, EvilClass(0))
+        self.assertRaises(IndexError, self.module.heappushpop, heap, 1)
+
+    def test_comparison_operator_modifiying_heap_two_heaps(self):
+
+        class h(int):
+            def __lt__(self, o):
+                list2.clear()
+                return NotImplemented
+
+        class g(int):
+            def __lt__(self, o):
+                list1.clear()
+                return NotImplemented
+
+        list1, list2 = [], []
+
+        self.module.heappush(list1, h(0))
+        self.module.heappush(list2, g(0))
+
+        self.assertRaises((IndexError, RuntimeError), self.module.heappush, list1, g(1))
+        self.assertRaises((IndexError, RuntimeError), self.module.heappush, list2, h(1))
 
 class TestErrorHandlingPython(TestErrorHandling, TestCase):
     module = py_heapq
diff --git a/Misc/NEWS.d/next/Core and Builtins/2020-01-22-15-53-37.bpo-39421.O3nG7u.rst b/Misc/NEWS.d/next/Core and Builtins/2020-01-22-15-53-37.bpo-39421.O3nG7u.rst
new file mode 100644
index 00000000000..bae008150ee
--- /dev/null
+++ b/Misc/NEWS.d/next/Core and Builtins/2020-01-22-15-53-37.bpo-39421.O3nG7u.rst	
@@ -0,0 +1,2 @@
+Fix possible crashes when operating with the functions in the :mod:`heapq`
+module and custom comparison operators.
diff --git a/Modules/_heapqmodule.c b/Modules/_heapqmodule.c
index a84cade3aaa..6bc18b5f82f 100644
--- a/Modules/_heapqmodule.c
+++ b/Modules/_heapqmodule.c
@@ -36,7 +36,11 @@ siftdown(PyListObject *heap, Py_ssize_t startpos, Py_ssize_t pos)
     while (pos > startpos) {
         parentpos = (pos - 1) >> 1;
         parent = arr[parentpos];
+        Py_INCREF(newitem);
+        Py_INCREF(parent);
         cmp = PyObject_RichCompareBool(newitem, parent, Py_LT);
+        Py_DECREF(parent);
+        Py_DECREF(newitem);
         if (cmp < 0)
             return -1;
         if (size != PyList_GET_SIZE(heap)) {
@@ -78,10 +82,13 @@ siftup(PyListObject *heap, Py_ssize_t pos)
         /* Set childpos to index of smaller child.   */
         childpos = 2*pos + 1;    /* leftmost child position  */
         if (childpos + 1 < endpos) {
-            cmp = PyObject_RichCompareBool(
-                arr[childpos],
-                arr[childpos + 1],
-                Py_LT);
+            PyObject* a = arr[childpos];
+            PyObject* b = arr[childpos + 1];
+            Py_INCREF(a);
+            Py_INCREF(b);
+            cmp = PyObject_RichCompareBool(a, b, Py_LT);
+            Py_DECREF(a);
+            Py_DECREF(b);
             if (cmp < 0)
                 return -1;
             childpos += ((unsigned)cmp ^ 1);   /* increment when cmp==0 */
@@ -264,7 +271,10 @@ _heapq_heappushpop_impl(PyObject *module, PyObject *heap, PyObject *item)
         return item;
     }
 
-    cmp = PyObject_RichCompareBool(PyList_GET_ITEM(heap, 0), item, Py_LT);
+    PyObject* top = PyList_GET_ITEM(heap, 0);
+    Py_INCREF(top);
+    cmp = PyObject_RichCompareBool(top, item, Py_LT);
+    Py_DECREF(top);
     if (cmp < 0)
         return NULL;
     if (cmp == 0) {
@@ -420,7 +430,11 @@ siftdown_max(PyListObject *heap, Py_ssize_t startpos, Py_ssize_t pos)
     while (pos > startpos) {
         parentpos = (pos - 1) >> 1;
         parent = arr[parentpos];
+        Py_INCREF(parent);
+        Py_INCREF(newitem);
         cmp = PyObject_RichCompareBool(parent, newitem, Py_LT);
+        Py_DECREF(parent);
+        Py_DECREF(newitem);
         if (cmp < 0)
             return -1;
         if (size != PyList_GET_SIZE(heap)) {
@@ -462,10 +476,13 @@ siftup_max(PyListObject *heap, Py_ssize_t pos)
         /* Set childpos to index of smaller child.   */
         childpos = 2*pos + 1;    /* leftmost child position  */
         if (childpos + 1 < endpos) {
-            cmp = PyObject_RichCompareBool(
-                arr[childpos + 1],
-                arr[childpos],
-                Py_LT);
+            PyObject* a = arr[childpos + 1];
+            PyObject* b = arr[childpos];
+            Py_INCREF(a);
+            Py_INCREF(b);
+            cmp = PyObject_RichCompareBool(a, b, Py_LT);
+            Py_DECREF(a);
+            Py_DECREF(b);
             if (cmp < 0)
                 return -1;
             childpos += ((unsigned)cmp ^ 1);   /* increment when cmp==0 */