diff --git a/azure-pipelines.yml b/azure-pipelines.yml
index d04351c270c4..2c3539229277 100644
--- a/azure-pipelines.yml
+++ b/azure-pipelines.yml
@@ -33,9 +33,7 @@ parameters:
   - name: "supportedPythonVersions"
     displayName: "All supported versions of Python"
     type: object
-    # TODO remove the explicit 3.11.0 pin and set it to 3.11. The pin is currently added
-    # since 3.11.1 breaks CI, see Qiskit/qiskit-terra#9291.
-    default: ["3.7", "3.8", "3.9", "3.10", "3.11.0"]
+    default: ["3.7", "3.8", "3.9", "3.10", "3.11"]
 
   - name: "minimumPythonVersion"
     displayName: "Minimum supported version of Python"
@@ -45,7 +43,7 @@ parameters:
   - name: "maximumPythonVersion"
     displayName: "Maximum supported version of Python"
     type: string
-    default: "3.11.0"
+    default: "3.11"
 
   - name: "minimumRustVersion"
     displayName: "Minimum supported version of Rust"
@@ -111,9 +109,7 @@ stages:
 
           - template: ".azure/test-macos.yml"
             parameters:
-              # TODO Manually setting this to exclude 3.11 completely, since 3.11.1 breaks
-              # (see Qiskit/qiskit-terra#9291) and 3.11.0 is not available on azure for MacOS.
-              pythonVersion: ["3.7", "3.8", "3.9", "3.10"]
+              pythonVersion: ${{ version }}
 
           - template: ".azure/test-windows.yml"
             parameters:
@@ -185,8 +181,7 @@ stages:
 
         - template: ".azure/test-macos.yml"
           parameters:
-            # TODO remove pin to 3.10 and reset to ${{ parameters.maximumPythonVersion }}
-            pythonVersion: "3.10"
+            pythonVersion: ${{ parameters.maximumPythonVersion }}
 
         - template: ".azure/test-windows.yml"
           parameters:
diff --git a/qiskit/utils/classtools.py b/qiskit/utils/classtools.py
index 71c17754bdad..1e58b1ad2b91 100644
--- a/qiskit/utils/classtools.py
+++ b/qiskit/utils/classtools.py
@@ -25,11 +25,6 @@
 _MAGIC_STATICMETHODS = {"__new__"}
 _MAGIC_CLASSMETHODS = {"__init_subclass__", "__prepare__"}
 
-# `type` itself has several methods (mostly dunders).  When we are wrapping those names, we need to
-# make sure that we don't interfere with `type.__getattribute__`'s handling that circumvents the
-# normal inheritance rules when appropriate.
-_TYPE_METHODS = set(dir(type))
-
 
 class _lift_to_method:  # pylint: disable=invalid-name
     """A decorator that ensures that an input callable object implements ``__get__``.  It is
@@ -146,16 +141,6 @@ def wrap_method(cls: Type, name: str, *, before: Callable = None, after: Callabl
     # The best time to apply decorators to methods is before they are bound (e.g. by using function
     # decorators during the class definition), but if we're making a class decorator, we can't do
     # that.  We need the actual definition of the method, so we have to dodge the normal output of
-    # `type.__getattribute__`, which evalutes descriptors if it finds them, unless the name we're
-    # looking for is defined on `type` itself.  In that case, we need the attribute getter to
-    # correctly return the underlying object, not the one that `type` defines for its own purposes.
-    attribute_getter = type.__getattribute__ if name in _TYPE_METHODS else object.__getattribute__
-    for cls_ in inspect.getmro(cls):
-        try:
-            method = attribute_getter(cls_, name)
-            break
-        except AttributeError:
-            pass
-    else:
-        raise ValueError(f"Method '{name}' is not defined for class '{cls.__name__}'")
+    # `type.__getattribute__`, which evalutes descriptors if it finds them.
+    method = inspect.getattr_static(cls, name)
     setattr(cls, name, _WrappedMethod(method, before, after))
diff --git a/releasenotes/notes/wrap-method-311-147d254d4b40e805.yaml b/releasenotes/notes/wrap-method-311-147d254d4b40e805.yaml
new file mode 100644
index 000000000000..146e01b903c5
--- /dev/null
+++ b/releasenotes/notes/wrap-method-311-147d254d4b40e805.yaml
@@ -0,0 +1,11 @@
+---
+fixes:
+  - |
+    Fixed handling of some ``classmethod``s by
+    :func:`~qiskit.utils.wrap_method` in Python 3.11. Previously, in Python
+    3.11, ``wrap_method`` would wrap the unbounded function associated with the
+    ``classmethod`` and then fail when invoked because the class object usually
+    bound to the ``classmethod`` was not passed to the function. Starting in
+    Python 3.11.1, this issue affected :class:`~qiskit.test.QiskitTestCase`,
+    preventing it from being imported by other test code. Fixed `#9291
+    <https://github.com/Qiskit/qiskit-terra/issues/9291>`__.
diff --git a/test/python/utils/test_classtools.py b/test/python/utils/test_classtools.py
index 272c3e0a6146..b373675ea569 100644
--- a/test/python/utils/test_classtools.py
+++ b/test/python/utils/test_classtools.py
@@ -529,5 +529,5 @@ def test_raises_on_invalid_name(self):
         class Dummy:
             pass
 
-        with self.assertRaisesRegex(ValueError, "Method 'bad' is not defined for class 'Dummy'"):
+        with self.assertRaisesRegex(AttributeError, "bad"):
             wrap_method(Dummy, "bad", before=lambda self: None)