diff --git a/Doc/c-api/module.rst b/Doc/c-api/module.rst
index c0351c8a6c72aa..4ab5afe7ca3147 100644
--- a/Doc/c-api/module.rst
+++ b/Doc/c-api/module.rst
@@ -123,6 +123,12 @@ Module Objects
       unencodable filenames, use :c:func:`PyModule_GetFilenameObject` instead.
 
 
+.. c:function:: int PyModule_Callable(PyObject *module)
+
+   Return true if *module* is a module object, and has a
+   :attr:`~object.__call__` property that is also callable.
+
+
 .. _initializing-modules:
 
 Initializing C modules
diff --git a/Include/moduleobject.h b/Include/moduleobject.h
index 555564ec73b4a2..1bb6adec66a0dd 100644
--- a/Include/moduleobject.h
+++ b/Include/moduleobject.h
@@ -34,6 +34,7 @@ PyAPI_FUNC(int) _PyModuleSpec_IsInitializing(PyObject *);
 #endif
 PyAPI_FUNC(PyModuleDef*) PyModule_GetDef(PyObject*);
 PyAPI_FUNC(void*) PyModule_GetState(PyObject*);
+PyAPI_FUNC(int) PyModule_Callable(PyObject *);
 
 #if !defined(Py_LIMITED_API) || Py_LIMITED_API+0 >= 0x03050000
 /* New in 3.5 */
diff --git a/Lib/test/callable_module_a.py b/Lib/test/callable_module_a.py
new file mode 100644
index 00000000000000..25e533091333fe
--- /dev/null
+++ b/Lib/test/callable_module_a.py
@@ -0,0 +1,6 @@
+"""
+PEP 713 Callable Module - def __call__()
+"""
+
+def __call__(value: int = 42, *, power: int = 1):
+    return value ** power
diff --git a/Lib/test/callable_module_b.py b/Lib/test/callable_module_b.py
new file mode 100644
index 00000000000000..1ccf9b4210b086
--- /dev/null
+++ b/Lib/test/callable_module_b.py
@@ -0,0 +1,8 @@
+"""
+PEP 713 Callable Module - __call__ = func
+"""
+
+def func(value: int = 42, *, power: int = 1):
+    return value ** power
+
+__call__ = func
diff --git a/Lib/test/callable_module_c.py b/Lib/test/callable_module_c.py
new file mode 100644
index 00000000000000..4e6e8a20584eaf
--- /dev/null
+++ b/Lib/test/callable_module_c.py
@@ -0,0 +1,11 @@
+"""
+PEP 713 Callable Module - __call__ = func
+"""
+
+def func(value: int = 42, *, power: int = 1):
+    return value ** power
+
+def __getattr__(name: str):
+    if name == "__call__":
+        return func
+    raise AttributeError
diff --git a/Lib/test/test_call.py b/Lib/test/test_call.py
index aab7b1580eaf35..dfc4f411b73ab0 100644
--- a/Lib/test/test_call.py
+++ b/Lib/test/test_call.py
@@ -945,5 +945,48 @@ def test_function_with_many_args(self):
                 self.assertEqual(l['f'](*range(N)), N//2)
 
 
+class TestPEP590(unittest.TestCase):
+    def test_sys_not_callable(self):
+        import sys
+
+        self.assertFalse(callable(sys))
+        self.assertFalse(hasattr(sys, "__call__"))
+        self.assertRaisesRegex(TypeError, "Module 'sys' is not callable", sys)
+
+    def test_sys_add_callable(self):
+        try:
+            def exponent(value=42, *, power=1):
+                return value ** power
+
+            sys.__call__ = exponent
+
+            self.assertTrue(callable(sys))
+            self.assertTrue(hasattr(sys, "__call__"))
+
+            self.assertEqual(sys(), 42)
+            self.assertEqual(sys(3), 3)
+            self.assertEqual(sys(power=2), 1764)
+
+        finally:
+            if hasattr(sys, "__call__"):
+                del sys.__call__
+
+    def test_callable_modules(self):
+        from . import callable_module_a, callable_module_b, callable_module_c
+
+        for name, module in (
+            ("via def __call__", callable_module_a),
+            ("via __call__ = func", callable_module_b),
+            ("via __getattr__", callable_module_c),
+        ):
+            with self.subTest(name):
+                self.assertTrue(callable(module))
+                self.assertTrue(hasattr(module, "__call__"))
+
+                self.assertEqual(module(), 42)
+                self.assertEqual(module(3), 3)
+                self.assertEqual(module(power=2), 1764)
+
+
 if __name__ == "__main__":
     unittest.main()
diff --git a/Objects/moduleobject.c b/Objects/moduleobject.c
index a0be19a3ca8ac8..e2bd1830e76151 100644
--- a/Objects/moduleobject.c
+++ b/Objects/moduleobject.c
@@ -743,7 +743,16 @@ module_getattro(PyModuleObject *m, PyObject *name)
 {
     PyObject *attr, *mod_name, *getattr;
     attr = PyObject_GenericGetAttr((PyObject *)m, name);
-    if (attr || !PyErr_ExceptionMatches(PyExc_AttributeError)) {
+    if (attr) {
+        // If the module does not explicitly set __call__, do not return
+        // the generic method-wrapper for tp_call in its place
+        if (!(Py_TYPE(attr) == &_PyMethodWrapper_Type &&
+              _PyUnicode_EqualToASCIIString(name, "__call__")))
+        {
+            return attr;
+        }
+    }
+    else if (!PyErr_ExceptionMatches(PyExc_AttributeError)) {
         return attr;
     }
     PyErr_Clear();
@@ -928,6 +937,38 @@ module_set_annotations(PyModuleObject *m, PyObject *value, void *Py_UNUSED(ignor
     return ret;
 }
 
+int
+PyModule_Callable(PyObject *mod)
+{
+    if (PyModule_CheckExact(mod)) {
+        PyObject *mod_call = PyObject_GetAttr(mod, &_Py_ID(__call__));
+        if (mod_call) {
+            return PyCallable_Check(mod_call);
+        }
+        else if (PyErr_ExceptionMatches(PyExc_AttributeError)) {
+            PyErr_Clear();
+            return 0;
+        }
+    }
+
+    return 0;
+}
+
+static PyObject *
+module_call(PyModuleObject *mod, PyObject *args, PyObject *kwargs)
+{
+    PyObject *callable = PyObject_GetAttr((PyObject *)mod, &_Py_ID(__call__));
+    if (callable == NULL) {
+        if (PyErr_ExceptionMatches(PyExc_AttributeError)) {
+            PyErr_Format(PyExc_TypeError,
+                        "Module '%.200s' is not callable",
+                        PyModule_GetName((PyObject *)mod));
+        }
+        return NULL;
+    }
+
+    return PyObject_Call(callable, args, kwargs);
+}
 
 static PyGetSetDef module_getsets[] = {
     {"__annotations__", (getter)module_get_annotations, (setter)module_set_annotations},
@@ -949,7 +990,7 @@ PyTypeObject PyModule_Type = {
     0,                                          /* tp_as_sequence */
     0,                                          /* tp_as_mapping */
     0,                                          /* tp_hash */
-    0,                                          /* tp_call */
+    (ternaryfunc)module_call,                   /* tp_call */
     0,                                          /* tp_str */
     (getattrofunc)module_getattro,              /* tp_getattro */
     PyObject_GenericSetAttr,                    /* tp_setattro */
diff --git a/Objects/object.c b/Objects/object.c
index 65c296e9340601..6996c1cd30bdca 100644
--- a/Objects/object.c
+++ b/Objects/object.c
@@ -1668,6 +1668,9 @@ PyCallable_Check(PyObject *x)
 {
     if (x == NULL)
         return 0;
+    if (PyModule_CheckExact(x)) {
+        return PyModule_Callable(x);
+    }
     return Py_TYPE(x)->tp_call != NULL;
 }