diff --git a/Lib/test/test_socket.py b/Lib/test/test_socket.py
index dc8a4a44f37..7b3914f30e5 100644
--- a/Lib/test/test_socket.py
+++ b/Lib/test/test_socket.py
@@ -5132,6 +5132,33 @@ class NonBlockingTCPTests(ThreadedTCPSocketTest):
         # send data: recv() will no longer block
         self.cli.sendall(MSG)
 
+    def testLargeTimeout(self):
+        # gh-126876: Check that a timeout larger than INT_MAX is replaced with
+        # INT_MAX in the poll() code path. The following assertion must not
+        # fail: assert(INT_MIN <= ms && ms <= INT_MAX).
+        large_timeout = _testcapi.INT_MAX + 1
+
+        # test recv() with large timeout
+        conn, addr = self.serv.accept()
+        self.addCleanup(conn.close)
+        try:
+            conn.settimeout(large_timeout)
+        except OverflowError:
+            # On Windows, settimeout() fails with OverflowError, whereas
+            # we want to test recv(). Just give up silently.
+            return
+        msg = conn.recv(len(MSG))
+
+    def _testLargeTimeout(self):
+        # test sendall() with large timeout
+        large_timeout = _testcapi.INT_MAX + 1
+        self.cli.connect((HOST, self.port))
+        try:
+            self.cli.settimeout(large_timeout)
+        except OverflowError:
+            return
+        self.cli.sendall(MSG)
+
 
 class FileObjectClassTestCase(SocketConnectedTest):
     """Unit tests for the object returned by socket.makefile()
diff --git a/Modules/socketmodule.c b/Modules/socketmodule.c
index 2764bd6e2b2..06be822c955 100644
--- a/Modules/socketmodule.c
+++ b/Modules/socketmodule.c
@@ -810,7 +810,9 @@ internal_select(PySocketSockObject *s, int writing, PyTime_t interval,
 
     /* s->sock_timeout is in seconds, timeout in ms */
     ms = _PyTime_AsMilliseconds(interval, _PyTime_ROUND_CEILING);
-    assert(ms <= INT_MAX);
+    if (ms > INT_MAX) {
+        ms = INT_MAX;
+    }
 
     /* On some OSes, typically BSD-based ones, the timeout parameter of the
        poll() syscall, when negative, must be exactly INFTIM, where defined,
@@ -822,6 +824,7 @@ internal_select(PySocketSockObject *s, int writing, PyTime_t interval,
         ms = -1;
 #endif
     }
+    assert(INT_MIN <= ms && ms <= INT_MAX);
 
     Py_BEGIN_ALLOW_THREADS;
     n = poll(&pollfd, 1, (int)ms);