diff --git a/Lib/test/test_genericalias.py b/Lib/test/test_genericalias.py
@@ -49,7 +49,10 @@
     ShareableList = None
 from os import DirEntry
 from re import Pattern, Match
-from types import GenericAlias, MappingProxyType, AsyncGeneratorType, CoroutineType, GeneratorType
+from types import (
+    AsyncGeneratorType, CoroutineType, GeneratorType, GenericAlias,
+    MappingProxyType,
+)
 from tempfile import TemporaryDirectory, SpooledTemporaryFile
 from urllib.parse import SplitResult, ParseResult
 from unittest.case import _AssertRaisesContext
@@ -98,11 +101,22 @@
 ]
 
 
+def generic_function[T]():
+    pass
+
+
+class GenericMethod:
+    def method[T](self):
+        pass
+
+
 class BaseTest(unittest.TestCase):
     """Test basics."""
     generic_types = [type, tuple, list, dict, frozendict,
                      set, frozenset, enumerate, memoryview,
                      slice,
+                     generic_function, GenericMethod().method, max,
+                     dict.fromkeys,
                      defaultdict, deque,
                      SequenceMatcher,
                      dircmp,
@@ -214,6 +228,26 @@ def test_no_chaining(self):
         with self.assertRaises(TypeError):
             t[int]
 
+    class Dummy:
+        pass
+
+    def test_callable_alias_does_not_set_orig_class(self):
+        dummy_type = self.Dummy
+
+        def function[T]():
+            result = dummy_type()
+            result.__orig_class__ = str
+            return result
+
+        class Class:
+            def method[T](self):
+                result = dummy_type()
+                result.__orig_class__ = str
+                return result
+
+        self.assertIs(function[int]().__orig_class__, str)
+        self.assertIs(Class().method[int]().__orig_class__, str)
+
     def test_generic_subclass(self):
         class MyList(list):
             pass

diff --git a/Objects/classobject.c b/Objects/classobject.c
@@ -310,13 +310,24 @@ method_descr_get(PyObject *meth, PyObject *obj, PyObject *cls)
     return meth;
 }
 
+static PyObject *
+method_getitem(PyObject *self, PyObject *item)
+{
+    return Py_GenericAlias(self, item);
+}
+
+static PyMappingMethods method_as_mapping = {
+    .mp_subscript = method_getitem,
+};
+
 PyTypeObject PyMethod_Type = {
     PyVarObject_HEAD_INIT(&PyType_Type, 0)
     .tp_name = "method",
     .tp_basicsize = sizeof(PyMethodObject),
     .tp_dealloc = method_dealloc,
     .tp_vectorcall_offset = offsetof(PyMethodObject, vectorcall),
     .tp_repr = method_repr,
+    .tp_as_mapping = &method_as_mapping,
     .tp_hash = method_hash,
     .tp_call = PyVectorcall_Call,
     .tp_getattro = method_getattro,

diff --git a/Objects/funcobject.c b/Objects/funcobject.c
@@ -1241,6 +1241,16 @@ func_repr(PyObject *self)
                                 op->func_qualname, op);
 }
 
+static PyObject *
+func_getitem(PyObject *self, PyObject *item)
+{
+    return Py_GenericAlias(self, item);
+}
+
+static PyMappingMethods func_as_mapping = {
+    .mp_subscript = func_getitem,
+};
+
 static int
 func_traverse(PyObject *self, visitproc visit, void *arg)
 {
@@ -1285,7 +1295,7 @@ PyTypeObject PyFunction_Type = {
     func_repr,                                  /* tp_repr */
     0,                                          /* tp_as_number */
     0,                                          /* tp_as_sequence */
-    0,                                          /* tp_as_mapping */
+    &func_as_mapping,                           /* tp_as_mapping */
     0,                                          /* tp_hash */
     PyVectorcall_Call,                          /* tp_call */
     0,                                          /* tp_str */

diff --git a/Objects/genericaliasobject.c b/Objects/genericaliasobject.c
@@ -643,8 +643,12 @@ ga_hash(PyObject *self)
 }
 
 static inline PyObject *
-set_orig_class(PyObject *obj, PyObject *self)
+set_orig_class(PyObject *obj, PyObject *self, PyObject *origin)
 {
+    if (PyFunction_Check(origin) || PyMethod_Check(origin) ||
+        PyCFunction_Check(origin)) {
+        return obj;
+    }
     if (obj != NULL) {
         if (PyObject_SetAttr(obj, &_Py_ID(__orig_class__), self) < 0) {
             if (!PyErr_ExceptionMatches(PyExc_AttributeError) &&
@@ -664,7 +668,7 @@ ga_call(PyObject *self, PyObject *args, PyObject *kwds)
 {
     gaobject *alias = (gaobject *)self;
     PyObject *obj = PyObject_Call(alias->origin, args, kwds);
-    return set_orig_class(obj, self);
+    return set_orig_class(obj, self, alias->origin);
 }
 
 static PyObject *
@@ -673,7 +677,7 @@ ga_vectorcall(PyObject *self, PyObject *const *args,
 {
     gaobject *alias = (gaobject *) self;
     PyObject *obj = PyObject_Vectorcall(alias->origin, args, nargsf, kwnames);
-    return set_orig_class(obj, self);
+    return set_orig_class(obj, self, alias->origin);
 }
 
 static const char* const attr_exceptions[] = {

diff --git a/Objects/methodobject.c b/Objects/methodobject.c
@@ -314,6 +314,16 @@ meth_repr(PyObject *self)
                                 m->m_self);
 }
 
+static PyObject *
+meth_getitem(PyObject *self, PyObject *item)
+{
+    return Py_GenericAlias(self, item);
+}
+
+static PyMappingMethods meth_as_mapping = {
+    .mp_subscript = meth_getitem,
+};
+
 static PyObject *
 meth_richcompare(PyObject *self, PyObject *other, int op)
 {
@@ -366,7 +376,7 @@ PyTypeObject PyCFunction_Type = {
     meth_repr,                                  /* tp_repr */
     0,                                          /* tp_as_number */
     0,                                          /* tp_as_sequence */
-    0,                                          /* tp_as_mapping */
+    &meth_as_mapping,                           /* tp_as_mapping */
     meth_hash,                                  /* tp_hash */
     cfunction_call,                             /* tp_call */
     0,                                          /* tp_str */