diff --git a/Lib/test/test_sys.py b/Lib/test/test_sys.py
--- a/Lib/test/test_sys.py
+++ b/Lib/test/test_sys.py
@@ -810,7 +810,7 @@ class SizeofTest(unittest.TestCase):
         # memoryview
         check(memoryview(b''), size('Pnin 2P2n2i5P 3cPn'))
         # module
-        check(unittest, size('PnP'))
+        check(unittest, size('PnPP'))
         # None
         check(None, size(''))
         # NotImplementedType
diff --git a/Objects/moduleobject.c b/Objects/moduleobject.c
--- a/Objects/moduleobject.c
+++ b/Objects/moduleobject.c
@@ -11,6 +11,7 @@ typedef struct {
     PyObject *md_dict;
     struct PyModuleDef *md_def;
     void *md_state;
+    PyObject *md_weaklist;
 } PyModuleObject;
 
 static PyMemberDef module_members[] = {
@@ -56,6 +57,7 @@ PyModule_NewObject(PyObject *name)
         return NULL;
     m->md_def = NULL;
     m->md_state = NULL;
+    m->md_weaklist = NULL;
     m->md_dict = PyDict_New();
     if (module_init_dict(m->md_dict, name, NULL) != 0)
         goto fail;
@@ -371,12 +373,11 @@ static void
 module_dealloc(PyModuleObject *m)
 {
     PyObject_GC_UnTrack(m);
+    if (m->md_weaklist != NULL)
+        PyObject_ClearWeakRefs((PyObject *) m);
     if (m->md_def && m->md_def->m_free)
         m->md_def->m_free(m);
-    if (m->md_dict != NULL) {
-        _PyModule_Clear((PyObject *)m);
-        Py_DECREF(m->md_dict);
-    }
+    Py_XDECREF(m->md_dict);
     if (m->md_state != NULL)
         PyMem_FREE(m->md_state);
     Py_TYPE(m)->tp_free((PyObject *)m);
@@ -522,7 +523,7 @@ PyTypeObject PyModule_Type = {
     (traverseproc)module_traverse,              /* tp_traverse */
     (inquiry)module_clear,                      /* tp_clear */
     0,                                          /* tp_richcompare */
-    0,                                          /* tp_weaklistoffset */
+    offsetof(PyModuleObject, md_weaklist),      /* tp_weaklistoffset */
     0,                                          /* tp_iter */
     0,                                          /* tp_iternext */
     module_methods,                             /* tp_methods */
diff --git a/Python/import.c b/Python/import.c
--- a/Python/import.c
+++ b/Python/import.c
@@ -323,6 +323,7 @@ PyImport_Cleanup(void)
     PyObject *key, *value, *dict;
     PyInterpreterState *interp = PyThreadState_GET()->interp;
     PyObject *modules = interp->modules;
+    PyObject *weaklist = NULL;
 
     if (modules == NULL)
         return; /* Already done */
@@ -360,12 +361,33 @@ PyImport_Cleanup(void)
         }
     }
 
+    /* We prepare a list which will receive (name, weakref) tuples of
+       modules when they are removed from sys.modules.  The name is used
+       for diagnosis messages (in verbose mode), while the weakref helps
+       detect those modules which have been held alive. */
+    weaklist = PyList_New(0);
+
+#define STORE_MODULE_WEAKREF(mod) \
+    if (weaklist != NULL) { \
+        PyObject *name = PyModule_GetNameObject(mod); \
+        PyObject *wr = PyWeakref_NewRef(mod, NULL); \
+        if (name && wr) { \
+            PyObject *tup = PyTuple_Pack(2, name, wr); \
+            PyList_Append(weaklist, tup); \
+            Py_XDECREF(tup); \
+        } \
+        Py_XDECREF(name); \
+        Py_XDECREF(wr); \
+        if (PyErr_Occurred()) \
+            PyErr_Clear(); \
+    }
+
     /* First, delete __main__ */
     value = PyDict_GetItemString(modules, "__main__");
     if (value != NULL && PyModule_Check(value)) {
         if (Py_VerboseFlag)
-            PySys_WriteStderr("# cleanup __main__\n");
-        _PyModule_Clear(value);
+            PySys_WriteStderr("# removing __main__\n");
+        STORE_MODULE_WEAKREF(value);
         PyDict_SetItemString(modules, "__main__", Py_None);
     }
 
@@ -396,8 +418,8 @@ PyImport_Cleanup(void)
                     continue;
                 if (Py_VerboseFlag)
                     PySys_FormatStderr(
-                        "# cleanup[1] %U\n", key);
-                _PyModule_Clear(value);
+                        "# cleanup[1] removing %U\n", key);
+                STORE_MODULE_WEAKREF(value);
                 PyDict_SetItem(modules, key, Py_None);
                 ndone++;
             }
@@ -411,8 +433,8 @@ PyImport_Cleanup(void)
             if (is_essential_module(key))
                 continue;
             if (Py_VerboseFlag)
-                PySys_FormatStderr("# cleanup[2] %U\n", key);
-            _PyModule_Clear(value);
+                PySys_FormatStderr("# cleanup[2] removing %U\n", key);
+            STORE_MODULE_WEAKREF(value);
             PyDict_SetItem(modules, key, Py_None);
         }
     }
@@ -430,17 +452,40 @@ PyImport_Cleanup(void)
     while (PyDict_Next(modules, &pos, &key, &value)) {
         if (PyUnicode_Check(key) && PyModule_Check(value)) {
             if (Py_VerboseFlag)
-                PySys_FormatStderr("# cleanup[3] %U\n", key);
-            _PyModule_Clear(value);
+                PySys_FormatStderr("# cleanup[3] removing %U\n", key);
+            STORE_MODULE_WEAKREF(value);
             PyDict_SetItem(modules, key, Py_None);
         }
     }
 
     /* Finally, clear and delete the modules directory */
     PyDict_Clear(modules);
-    _PyGC_CollectNoFail();
     interp->modules = NULL;
     Py_DECREF(modules);
+
+    /* If there are any modules left alive, clear their globals to minimize
+       potential leaks. */
+    if (weaklist != NULL) {
+        Py_ssize_t i, n;
+        n = PyList_GET_SIZE(weaklist);
+        for (i = 0; i < n; i++) {
+            PyObject *tup = PyList_GET_ITEM(weaklist, i);
+            PyObject *mod = PyWeakref_GET_OBJECT(PyTuple_GET_ITEM(tup, 1));
+            if (mod == Py_None)
+                continue;
+            Py_INCREF(mod);
+            assert(PyModule_Check(mod));
+            if (Py_VerboseFlag)
+                PySys_FormatStderr("# cleanup[4] wiping %U\n",
+                                   PyTuple_GET_ITEM(tup, 0));
+            _PyModule_Clear(mod);
+            Py_DECREF(mod);
+        }
+        Py_DECREF(weaklist);
+    }
+    _PyGC_CollectNoFail();
+
+#undef STORE_MODULE_WEAKREF
 }