diff --git a/Lib/importlib/_bootstrap.py b/Lib/importlib/_bootstrap.py
--- a/Lib/importlib/_bootstrap.py
+++ b/Lib/importlib/_bootstrap.py
@@ -780,6 +780,9 @@ class _FileFinder:
             mod_filename = tail_module + suffix
             full_path = _path_join(self.path, mod_filename)
             if _path_isfile(full_path) and _case_ok(self.path, mod_filename):
+                if "module" in suffix:
+                    msg = "deprecated module file name format: {!r}"
+                    _warnings.warn(msg.format(mod_filename), DeprecationWarning)
                 return loader(fullname, full_path)
         return None
 
diff --git a/Lib/importlib/test/extension/test_finder.py b/Lib/importlib/test/extension/test_finder.py
--- a/Lib/importlib/test/extension/test_finder.py
+++ b/Lib/importlib/test/extension/test_finder.py
@@ -2,8 +2,13 @@ from importlib import _bootstrap
 from .. import abc
 from . import util
 
+import imp
+import os
 import unittest
 
+from test.support import TESTFN, unlink
+
+
 class FinderTests(abc.FinderTests):
 
     """Test the finder for extension modules."""
@@ -37,6 +42,26 @@ class FinderTests(abc.FinderTests):
 
     # XXX Raise an exception if someone tries to use the 'path' argument?
 
+    def test_deprecated_suffixes(self):
+        # Issue #14040: deprecated C extension suffixes
+        suffixes = [suffix for suffix, _, kind in imp.get_suffixes()
+                    if kind == imp.C_EXTENSION and 'module' in suffix]
+        if not suffixes:
+            self.skipTest("no deprecated suffixes to test for")
+        importer = _bootstrap._FileFinder(os.curdir,
+                                          _bootstrap._ExtensionFinderDetails())
+        for suffix in suffixes:
+            module_filename = TESTFN + suffix
+            with open(module_filename, 'wb') as f:
+                pass
+            try:
+                with self.assertWarns(DeprecationWarning) as cm:
+                    imp.find_module(TESTFN, [os.curdir])
+                self.assertIn(module_filename, str(cm.warning))
+            finally:
+                unlink(module_filename)
+
+
 
 def test_main():
     from test.support import run_unittest
diff --git a/Lib/test/test_import.py b/Lib/test/test_import.py
--- a/Lib/test/test_import.py
+++ b/Lib/test/test_import.py
@@ -331,6 +331,23 @@ class ImportTests(unittest.TestCase):
             del sys.path[0]
             remove_files(TESTFN)
 
+    def test_deprecated_suffixes(self):
+        # Issue #14040: deprecated C extension suffixes
+        suffixes = [suffix for suffix, _, kind in imp.get_suffixes()
+                    if kind == imp.C_EXTENSION and 'module' in suffix]
+        if not suffixes:
+            self.skipTest("no deprecated suffixes to test for")
+        for suffix in suffixes:
+            module_filename = TESTFN + suffix
+            with open(module_filename, 'wb') as f:
+                pass
+            try:
+                with self.assertWarns(DeprecationWarning) as cm:
+                    imp.find_module(TESTFN, [os.curdir])
+                self.assertIn(module_filename, str(cm.warning))
+            finally:
+                unlink(module_filename)
+
 
 class PycRewritingTests(unittest.TestCase):
     # Test that the `co_filename` attribute on code objects always points
diff --git a/Python/dynload_aix.c b/Python/dynload_aix.c
--- a/Python/dynload_aix.c
+++ b/Python/dynload_aix.c
@@ -27,8 +27,8 @@ typedef struct Module {
 } Module, *ModulePtr;
 
 const struct filedescr _PyImport_DynLoadFiletab[] = {
-    {".so", "rb", C_EXTENSION},
-    {"module.so", "rb", C_EXTENSION},
+    {".so", "rb", C_EXTENSION, 0},
+    {"module.so", "rb", C_EXTENSION, 1},
     {0, 0}
 };
 
diff --git a/Python/dynload_dl.c b/Python/dynload_dl.c
--- a/Python/dynload_dl.c
+++ b/Python/dynload_dl.c
@@ -10,8 +10,8 @@
 extern char *Py_GetProgramName(void);
 
 const struct filedescr _PyImport_DynLoadFiletab[] = {
-    {".o", "rb", C_EXTENSION},
-    {"module.o", "rb", C_EXTENSION},
+    {".o", "rb", C_EXTENSION, 0},
+    {"module.o", "rb", C_EXTENSION, 1},
     {0, 0}
 };
 
diff --git a/Python/dynload_hpux.c b/Python/dynload_hpux.c
--- a/Python/dynload_hpux.c
+++ b/Python/dynload_hpux.c
@@ -14,8 +14,8 @@
 #endif
 
 const struct filedescr _PyImport_DynLoadFiletab[] = {
-    {SHLIB_EXT, "rb", C_EXTENSION},
-    {"module"SHLIB_EXT, "rb", C_EXTENSION},
+    {SHLIB_EXT, "rb", C_EXTENSION, 0},
+    {"module"SHLIB_EXT, "rb", C_EXTENSION, 1},
     {0, 0}
 };
 
diff --git a/Python/dynload_next.c b/Python/dynload_next.c
--- a/Python/dynload_next.c
+++ b/Python/dynload_next.c
@@ -9,8 +9,8 @@
 #include <mach-o/dyld.h>
 
 const struct filedescr _PyImport_DynLoadFiletab[] = {
-    {".so", "rb", C_EXTENSION},
-    {"module.so", "rb", C_EXTENSION},
+    {".so", "rb", C_EXTENSION, 0},
+    {"module.so", "rb", C_EXTENSION, 1},
     {0, 0}
 };
 
diff --git a/Python/dynload_os2.c b/Python/dynload_os2.c
--- a/Python/dynload_os2.c
+++ b/Python/dynload_os2.c
@@ -10,8 +10,8 @@
 
 
 const struct filedescr _PyImport_DynLoadFiletab[] = {
-    {".pyd", "rb", C_EXTENSION},
-    {".dll", "rb", C_EXTENSION},
+    {".pyd", "rb", C_EXTENSION, 0},
+    {".dll", "rb", C_EXTENSION, 0},
     {0, 0}
 };
 
diff --git a/Python/dynload_shlib.c b/Python/dynload_shlib.c
--- a/Python/dynload_shlib.c
+++ b/Python/dynload_shlib.c
@@ -38,25 +38,25 @@
 
 const struct filedescr _PyImport_DynLoadFiletab[] = {
 #ifdef __CYGWIN__
-    {".dll", "rb", C_EXTENSION},
-    {"module.dll", "rb", C_EXTENSION},
+    {".dll", "rb", C_EXTENSION, 0},
+    {"module.dll", "rb", C_EXTENSION, 1},
 #else  /* !__CYGWIN__ */
 #if defined(PYOS_OS2) && defined(PYCC_GCC)
-    {".pyd", "rb", C_EXTENSION},
-    {".dll", "rb", C_EXTENSION},
+    {".pyd", "rb", C_EXTENSION, 0},
+    {".dll", "rb", C_EXTENSION, 1},
 #else  /* !(defined(PYOS_OS2) && defined(PYCC_GCC)) */
 #ifdef __VMS
-    {".exe", "rb", C_EXTENSION},
-    {".EXE", "rb", C_EXTENSION},
-    {"module.exe", "rb", C_EXTENSION},
-    {"MODULE.EXE", "rb", C_EXTENSION},
+    {".exe", "rb", C_EXTENSION, 0},
+    {".EXE", "rb", C_EXTENSION, 0},
+    {"module.exe", "rb", C_EXTENSION, 0},
+    {"MODULE.EXE", "rb", C_EXTENSION, 0},
 #else  /* !__VMS */
-    {"." SOABI ".so", "rb", C_EXTENSION},
-    {"module." SOABI ".so", "rb", C_EXTENSION},
-    {".abi" PYTHON_ABI_STRING ".so", "rb", C_EXTENSION},
-    {"module.abi" PYTHON_ABI_STRING ".so", "rb", C_EXTENSION},
-    {".so", "rb", C_EXTENSION},
-    {"module.so", "rb", C_EXTENSION},
+    {"." SOABI ".so", "rb", C_EXTENSION, 0},
+    {"module." SOABI ".so", "rb", C_EXTENSION, 1},
+    {".abi" PYTHON_ABI_STRING ".so", "rb", C_EXTENSION, 0},
+    {"module.abi" PYTHON_ABI_STRING ".so", "rb", C_EXTENSION, 1},
+    {".so", "rb", C_EXTENSION, 0},
+    {"module.so", "rb", C_EXTENSION, 1},
 #endif  /* __VMS */
 #endif  /* defined(PYOS_OS2) && defined(PYCC_GCC) */
 #endif  /* __CYGWIN__ */
diff --git a/Python/dynload_win.c b/Python/dynload_win.c
--- a/Python/dynload_win.c
+++ b/Python/dynload_win.c
@@ -17,9 +17,9 @@ void _Py_DeactivateActCtx(ULONG_PTR cook
 
 const struct filedescr _PyImport_DynLoadFiletab[] = {
 #ifdef _DEBUG
-    {"_d.pyd", "rb", C_EXTENSION},
+    {"_d.pyd", "rb", C_EXTENSION, 0},
 #else
-    {".pyd", "rb", C_EXTENSION},
+    {".pyd", "rb", C_EXTENSION, 0},
 #endif
     {0, 0}
 };
diff --git a/Python/import.c b/Python/import.c
--- a/Python/import.c
+++ b/Python/import.c
@@ -1994,6 +1994,14 @@ find_module_path_list(PyObject *fullname
             }
             if (match) {
                 Py_DECREF(prefix);
+                if (fdp->deprecated &&
+                    PyErr_WarnFormat(PyExc_DeprecationWarning, 1,
+                                     "deprecated module file name format: %R",
+                                     filename)) {
+                    Py_DECREF(filename);
+                    fclose(fp);
+                    return NULL;
+                }
                 *p_path = filename;
                 *p_fp = fp;
                 return fdp;
diff --git a/Python/importdl.h b/Python/importdl.h
--- a/Python/importdl.h
+++ b/Python/importdl.h
@@ -24,6 +24,7 @@ struct filedescr {
     char *suffix;
     char *mode;
     enum filetype type;
+    int deprecated;
 };
 extern struct filedescr * _PyImport_Filetab;
 extern const struct filedescr _PyImport_DynLoadFiletab[];