diff --git a/easybuild/easyblocks/generic/pythonpackage.py b/easybuild/easyblocks/generic/pythonpackage.py
index ec939223037..ab41b7505fc 100644
--- a/easybuild/easyblocks/generic/pythonpackage.py
+++ b/easybuild/easyblocks/generic/pythonpackage.py
@@ -110,7 +110,7 @@ def check_python_cmd(python_cmd):
                 log.debug(f"Python command '{python_cmd}' not available through $PATH")
                 return False
 
-        pyver = det_python_version(python_cmd)
+        pyver = LooseVersion(det_python_version(python_cmd))
 
         if req_maj_ver is not None:
             if req_min_ver is None:
@@ -119,26 +119,28 @@ def check_python_cmd(python_cmd):
                 req_majmin_ver = '%s.%s' % (req_maj_ver, req_min_ver)
 
             # (strict) check for major version
-            maj_ver = pyver.split('.')[0]
-            if maj_ver != str(req_maj_ver):
+            maj_ver = pyver.version[0]
+            if maj_ver != req_maj_ver:
                 log.debug(f"Major Python version does not match: {maj_ver} vs {req_maj_ver}")
                 return False
 
             # check for minimal minor version
-            if LooseVersion(pyver) < LooseVersion(req_majmin_ver):
+            if pyver < req_majmin_ver:
                 log.debug(f"Minimal requirement for minor Python version not satisfied: {pyver} vs {req_majmin_ver}")
                 return False
 
         if max_py_majver is not None:
             if max_py_minver is None:
-                max_majmin_ver = '%s.0' % max_py_majver
+                max_ver = int(max_py_majver)
+                tested_pyver = pyver.version[0]
             else:
-                max_majmin_ver = '%s.%s' % (max_py_majver, max_py_minver)
+                max_ver = LooseVersion('%s.%s' % (max_py_majver, max_py_minver))
+                # Make sure we test only until the minor version, because 3.9.3 > 3.9 but we want to allow this
+                tested_pyver = '.'.join(str(v) for v in pyver.version[:2])
 
-            if LooseVersion(pyver) > LooseVersion(max_majmin_ver):
-                log.debug("Python version (%s) on the system is newer than the maximum supported "
-                          "Python version specified in the easyconfig (%s)",
-                          pyver, max_majmin_ver)
+            if tested_pyver > max_ver:
+                log.debug(f"Python version ({pyver}) on the system is newer than the maximum supported "
+                          f"Python version specified in the easyconfig ({max_ver})")
                 return False
 
         # all check passed
@@ -194,8 +196,9 @@ def find_python_cmd(log, req_py_majver, req_py_minver, max_py_majver, max_py_min
         # if no Python version requirements are specified,
         # use major/minor version of Python being used in this EasyBuild session
         if req_py_majver is None:
+            if req_py_minver is not None:
+                raise EasyBuildError("'req_py_majver' must be specified when 'req_py_minver' is set!")
             req_py_majver = sys.version_info[0]
-        if req_py_minver is None:
             req_py_minver = sys.version_info[1]
         # if using system Python, go hunting for a 'python' command that satisfies the requirements
         python = pick_python_cmd(req_maj_ver=req_py_majver, req_min_ver=req_py_minver,
diff --git a/test/easyblocks/module.py b/test/easyblocks/module.py
index a0437e495b8..d30736a63dd 100644
--- a/test/easyblocks/module.py
+++ b/test/easyblocks/module.py
@@ -233,12 +233,20 @@ def test_pythonpackage_pick_python_cmd(self):
         tmpdir = tempfile.mkdtemp()
         for cmd in ('python2', 'python2.6'):
             install_fake_command(cmd, "#!/bin/bash\n echo 2.6.4", tmpdir)
-        self.assertTrue(pick_python_cmd() is not None)
-        self.assertTrue(pick_python_cmd(3) is not None)
-        self.assertTrue(pick_python_cmd(3, 6) is not None)
-        self.assertTrue(pick_python_cmd(123, 456) is None)
-        self.assertTrue(pick_python_cmd(3, 6, 123, 456) is not None)
-        self.assertTrue(pick_python_cmd(2, 6, 1, 1) is None)
+        self.assertIsNotNone(pick_python_cmd())
+        self.assertIsNotNone(pick_python_cmd(3))
+        self.assertIsNotNone(pick_python_cmd(3, 6))
+        self.assertIsNone(pick_python_cmd(123, 456))
+        self.assertIsNotNone(pick_python_cmd(2, 6, 123, 456))
+        self.assertIsNotNone(pick_python_cmd(2, 6, 2))
+        self.assertIsNone(pick_python_cmd(2, 6, 1, 1))
+        maj_ver, min_ver = sys.version_info[0:2]
+        self.assertIsNotNone(pick_python_cmd(maj_ver, min_ver))
+        self.assertIsNotNone(pick_python_cmd(maj_ver, min_ver, max_py_majver=maj_ver))
+        self.assertIsNotNone(pick_python_cmd(maj_ver, min_ver, max_py_majver=maj_ver, max_py_minver=min_ver))
+        self.assertIsNotNone(pick_python_cmd(maj_ver, min_ver, max_py_majver=maj_ver, max_py_minver=min_ver + 1))
+        self.assertIsNone(pick_python_cmd(maj_ver, min_ver, max_py_majver=maj_ver - 1))
+        self.assertIsNone(pick_python_cmd(maj_ver, min_ver, max_py_majver=maj_ver, max_py_minver=min_ver - 1))
 
 
 def template_module_only_test(self, easyblock, name, version='1.3.2', extra_txt='', tmpdir=None):