diff --git a/build/pkgs/python3/spkg-configure.m4 b/build/pkgs/python3/spkg-configure.m4
index df740192aca..5e3481a6ee1 100644
--- a/build/pkgs/python3/spkg-configure.m4
+++ b/build/pkgs/python3/spkg-configure.m4
@@ -1,7 +1,8 @@
 SAGE_SPKG_CONFIGURE([python3], [
    m4_pushdef([MIN_VERSION],               [3.7.0])
    m4_pushdef([MIN_NONDEPRECATED_VERSION], [3.7.0])
-   m4_pushdef([LT_VERSION],                [3.10.0])
+   m4_pushdef([LT_STABLE_VERSION],         [3.10.0])
+   m4_pushdef([LT_VERSION],                [3.11.0])
    AC_ARG_WITH([python],
                [AS_HELP_STRING([--with-python=PYTHON3],
                                [Python 3 executable to use for the Sage venv; default: python3])])
@@ -114,6 +115,10 @@ SAGE_SPKG_CONFIGURE([python3], [
             AC_MSG_NOTICE([deprecation notice: Support for system python < MIN_NONDEPRECATED_VERSION is deprecated
 and will be removed in the next development cycle.  Consider using a newer version of Python
 that may be available on your system or can be installed using the system package manager.
+To build Sage with a different system python, use ./configure --with-python=/path/to/python])
+        ])
+        AX_COMPARE_VERSION([$python3_version], [ge], LT_STABLE_VERSION, [
+            AC_MSG_WARN([Support for system python >= LT_STABLE_VERSION is experimental.
 To build Sage with a different system python, use ./configure --with-python=/path/to/python])
         ])
     ])
diff --git a/pkgs/sage-setup/setup.cfg b/pkgs/sage-setup/setup.cfg
index a1710d407e2..197fc6791dc 100644
--- a/pkgs/sage-setup/setup.cfg
+++ b/pkgs/sage-setup/setup.cfg
@@ -31,7 +31,7 @@ packages =
     sage_setup.autogen.interpreters.specs
     sage_setup.command
 
-python_requires = >=3.7, <3.10
+python_requires = >=3.7, <3.11
 
 install_requires =
     pkgconfig
diff --git a/src/sage/cpython/atexit.pyx b/src/sage/cpython/atexit.pyx
index 5b0c6d10b2c..3ba391b4cdd 100644
--- a/src/sage/cpython/atexit.pyx
+++ b/src/sage/cpython/atexit.pyx
@@ -144,43 +144,63 @@ cdef class restore_atexit:
 
 from cpython.ref cimport PyObject
 
-# Internal structures defined in the CPython source in
-# Modules/atexitmodule.c and subject to (but unlikely to) change.  Watch
-# https://bugs.python.org/issue32082 for a request to (eventually)
-# re-expose more of the atexit module's internals to Python
-ctypedef struct atexit_callback:
-    PyObject* func
-    PyObject* args
-    PyObject* kwargs
-
-
-ctypedef struct atexitmodule_state:
-    atexit_callback** atexit_callbacks
-    int ncallbacks
-    int callback_len
-
-
-cdef extern from "Python.h":
-    void* PyModule_GetState(object module)
-
+# Implement "_atexit_callbacks()" for each supported python version
+cdef extern from *:
+    """
+    #if PY_VERSION_HEX >= 0x030a0000
+    /********** Python 3.10 **********/
+    #define Py_BUILD_CORE
+    #undef _PyGC_FINALIZED
+    #include "internal/pycore_interp.h"
+    #include "internal/pycore_pystate.h"
+    static atexit_callback ** _atexit_callbacks(PyObject *self) {
+        PyInterpreterState *interp = _PyInterpreterState_GET();
+        struct atexit_state state = interp->atexit;
+        return state.callbacks;
+    }
+    #else
+    /********** Python < 3.10 **********/
+    /* Internal structures defined in the CPython source in
+     * Modules/atexitmodule.c and subject to (but unlikely to) change.  Watch
+     * https://bugs.python.org/issue32082 for a request to (eventually)
+     * re-expose more of the atexit module's internals to Python
+     * typedef struct
+     */
+    typedef struct {
+        PyObject *func;
+        PyObject *args;
+        PyObject *kwargs;
+    } atexit_callback;
+    typedef struct {
+        atexit_callback **atexit_callbacks;
+        int ncallbacks;
+        int callback_len;
+    } atexitmodule_state;
+    static atexit_callback ** _atexit_callbacks(PyObject *self) {
+        atexitmodule_state *state = PyModule_GetState(self);
+        return state->atexit_callbacks;
+    }
+    #endif
+    """
+    ctypedef struct atexit_callback:
+        PyObject* func
+        PyObject* args
+        PyObject* kwargs
+    atexit_callback** _atexit_callbacks(object module)
 
 def _get_exithandlers():
     """Return list of exit handlers registered with the atexit module."""
-    cdef atexitmodule_state* state
+    cdef atexit_callback ** callbacks
     cdef atexit_callback callback
     cdef list exithandlers
     cdef int idx
     cdef object kwargs
 
-    state = <atexitmodule_state*>PyModule_GetState(atexit)
-
-    if not state:
-        raise RuntimeError("atexit module state missing or corrupt")
-
     exithandlers = []
+    callbacks = _atexit_callbacks(atexit)
 
-    for idx in range(state.ncallbacks):
-        callback = state.atexit_callbacks[idx][0]
+    for idx in range(atexit._ncallbacks()):
+        callback = callbacks[idx][0]
         if callback.kwargs:
             kwargs = <object>callback.kwargs
         else:
@@ -207,4 +227,5 @@ def _set_exithandlers(exithandlers):
 
 def _clear_exithandlers():
     """Clear the atexit module of all registered exit handlers."""
+
     atexit._clear()
diff --git a/src/sage_docbuild/utils.py b/src/sage_docbuild/utils.py
index 956d703fd05..3e4793e5ca4 100644
--- a/src/sage_docbuild/utils.py
+++ b/src/sage_docbuild/utils.py
@@ -79,7 +79,7 @@ def build_many(target, args, processes=None):
         ....:         # Task 4 is a poison pill
         ....:         1 / 0
         ....:     else:
-        ....:         time.sleep(0.5)
+        ....:         time.sleep(float(0.5))
         ....:         print('Processed task %s' % N)
         ....:
 
@@ -104,7 +104,7 @@ def build_many(target, args, processes=None):
         ....:         # Task 4 is a poison pill
         ....:         os.kill(os.getpid(), signal.SIGKILL)
         ....:     else:
-        ....:         time.sleep(0.5)
+        ....:         time.sleep(float(0.5))
         ....:         print('Processed task %s' % N)
         ....:
         sage: build_many(target, range(8), processes=8)
diff --git a/src/setup.cfg.m4 b/src/setup.cfg.m4
index 5c866c8c32b..d2012aa7f9f 100644
--- a/src/setup.cfg.m4
+++ b/src/setup.cfg.m4
@@ -26,7 +26,7 @@ classifiers =
     Topic :: Scientific/Engineering :: Mathematics
 
 [options]
-python_requires = >=3.7, <3.10
+python_requires = >=3.7, <3.11
 install_requires =
     sage_conf
     esyscmd(`sage-get-system-packages install-requires \