diff --git a/m4/ax_python.m4 b/m4/ax_python.m4
index d293da525..9f43ea0ab 100644
--- a/m4/ax_python.m4
+++ b/m4/ax_python.m4
@@ -3,7 +3,7 @@ dnl 2019 David Lamparter for NetDEF, Inc.
 dnl SPDX-License-Identifier: GPL-2.0-or-later
 
 dnl the _ at the beginning will be cut off (to support the empty version string)
-m4_define_default([_FRR_PY_VERS], [_3 _ _2 _3.7 _3.6 _3.5 _3.4 _3.3 _3.2 _2.7])
+m4_define_default([_FRR_PY_VERS], [_3 _3.10 _3.9 _3.8 _3.7 _3.6 _3.5 _3.4 _3.3 _3.2 _ _2 _2.7])
 
 dnl check basic interpreter properties (py2/py3)
 dnl doubles as simple check whether the interpreter actually works
@@ -186,7 +186,8 @@ AC_REQUIRE([PKG_PROG_PKG_CONFIG])dnl
       AC_MSG_RESULT([yes])
 
       PYTHON_CFLAGS="`\"$pycfg\" --includes`"
-      if test x"${py_ver}" == x"3.8" || test x"{py_ver}" == x"3.9"; then
+      minor_ver=${py_ver#*\.}
+      if test $((minor_ver)) > 7; then
         PYTHON_LIBS="`\"$pycfg\" --ldflags --embed`"
       else
         PYTHON_LIBS="`\"$pycfg\" --ldflags`"