diff --git a/google/auth/transport/requests.py b/google/auth/transport/requests.py
index 1caec0b..ce78e63 100644
--- a/google/auth/transport/requests.py
+++ b/google/auth/transport/requests.py
@@ -239,18 +239,38 @@
         # credentials.refresh).
         self._auth_request = auth_request
 
-    def request(self, method, url, data=None, headers=None, timeout=None, **kwargs):
+    def request(
+        self,
+        method,
+        url,
+        data=None,
+        headers=None,
+        max_allowed_time=None,
+        timeout=None,
+        **kwargs
+    ):
         """Implementation of Requests' request.
 
         Args:
-            timeout (Optional[Union[float, Tuple[float, float]]]): The number
-                of seconds to wait before raising a ``Timeout`` exception. If
-                multiple requests are made under the hood, ``timeout`` is
-                interpreted as the approximate total time of **all** requests.
+            timeout (Optional[Union[float, Tuple[float, float]]]):
+                The amount of time in seconds to wait for the server response
+                with each individual request.
 
-                If passed as a tuple ``(connect_timeout, read_timeout)``, the
-                smaller of the values is taken as the total allowed time across
-                all requests.
+                Can also be passed as a tuple (connect_timeout, read_timeout).
+                See :meth:`requests.Session.request` documentation for details.
+
+            max_allowed_time (Optional[float]):
+                If the method runs longer than this, a ``Timeout`` exception is
+                automatically raised. Unlike the ``timeout` parameter, this
+                value applies to the total method execution time, even if
+                multiple requests are made under the hood.
+
+                Mind that it is not guaranteed that the timeout error is raised
+                at ``max_allowed_time`. It might take longer, for example, if
+                an underlying request takes a lot of time, but the request
+                itself does not timeout, e.g. if a large file is being
+                transmitted. The timout error will be raised after such
+                request completes.
         """
         # pylint: disable=arguments-differ
         # Requests has a ton of arguments to request, but only two
@@ -273,11 +293,13 @@
             else functools.partial(self._auth_request, timeout=timeout)
         )
 
-        with TimeoutGuard(timeout) as guard:
-            self.credentials.before_request(auth_request, method, url, request_headers)
-        timeout = guard.remaining_timeout
+        remaining_time = max_allowed_time
 
-        with TimeoutGuard(timeout) as guard:
+        with TimeoutGuard(remaining_time) as guard:
+            self.credentials.before_request(auth_request, method, url, request_headers)
+        remaining_time = guard.remaining_timeout
+
+        with TimeoutGuard(remaining_time) as guard:
             response = super(AuthorizedSession, self).request(
                 method,
                 url,
@@ -286,7 +308,7 @@
                 timeout=timeout,
                 **kwargs
             )
-        timeout = guard.remaining_timeout
+        remaining_time = guard.remaining_timeout
 
         # If the response indicated that the credentials needed to be
         # refreshed, then refresh the credentials and re-attempt the
@@ -305,14 +327,6 @@
                 self._max_refresh_attempts,
             )
 
-            if self._refresh_timeout is not None:
-                if timeout is None:
-                    timeout = self._refresh_timeout
-                elif isinstance(timeout, numbers.Number):
-                    timeout = min(timeout, self._refresh_timeout)
-                else:
-                    timeout = tuple(min(x, self._refresh_timeout) for x in timeout)
-
             # Do not apply the timeout unconditionally in order to not override the
             # _auth_request's default timeout.
             auth_request = (
@@ -321,17 +335,18 @@
                 else functools.partial(self._auth_request, timeout=timeout)
             )
 
-            with TimeoutGuard(timeout) as guard:
+            with TimeoutGuard(remaining_time) as guard:
                 self.credentials.refresh(auth_request)
-            timeout = guard.remaining_timeout
+            remaining_time = guard.remaining_timeout
 
             # Recurse. Pass in the original headers, not our modified set, but
-            # do pass the adjusted timeout (i.e. the remaining time).
+            # do pass the adjusted max allowed time (i.e. the remaining total time).
             return self.request(
                 method,
                 url,
                 data=data,
                 headers=headers,
+                max_allowed_time=remaining_time,
                 timeout=timeout,
                 _credential_refresh_attempt=_credential_refresh_attempt + 1,
                 **kwargs
diff --git a/tests/transport/test_requests.py b/tests/transport/test_requests.py
index 0026974..8f73d4b 100644
--- a/tests/transport/test_requests.py
+++ b/tests/transport/test_requests.py
@@ -220,7 +220,25 @@
         assert adapter.requests[1].url == self.TEST_URL
         assert adapter.requests[1].headers["authorization"] == "token1"
 
-    def test_request_timeout(self, frozen_time):
+    def test_request_max_allowed_time_timeout_error(self, frozen_time):
+        tick_one_second = functools.partial(frozen_time.tick, delta=1.0)
+
+        credentials = mock.Mock(
+            wraps=TimeTickCredentialsStub(time_tick=tick_one_second)
+        )
+        adapter = TimeTickAdapterStub(
+            time_tick=tick_one_second, responses=[make_response(status=http_client.OK)]
+        )
+
+        authed_session = google.auth.transport.requests.AuthorizedSession(credentials)
+        authed_session.mount(self.TEST_URL, adapter)
+
+        # Because a request takes a full mocked second, max_allowed_time shorter
+        # than that will cause a timeout error.
+        with pytest.raises(requests.exceptions.Timeout):
+            authed_session.request("GET", self.TEST_URL, max_allowed_time=0.9)
+
+    def test_request_max_allowed_time_w_transport_timeout_no_error(self, frozen_time):
         tick_one_second = functools.partial(frozen_time.tick, delta=1.0)
 
         credentials = mock.Mock(
@@ -237,12 +255,12 @@
         authed_session = google.auth.transport.requests.AuthorizedSession(credentials)
         authed_session.mount(self.TEST_URL, adapter)
 
-        # Because at least two requests have to be made, and each takes one
-        # second, the total timeout specified will be exceeded.
-        with pytest.raises(requests.exceptions.Timeout):
-            authed_session.request("GET", self.TEST_URL, timeout=1.9)
+        # A short configured transport timeout does not affect max_allowed_time.
+        # The latter is not adjusted to it and is only concerned with the actual
+        # execution time. The call below should thus not raise a timeout error.
+        authed_session.request("GET", self.TEST_URL, timeout=0.5, max_allowed_time=3.1)
 
-    def test_request_timeout_w_refresh_timeout(self, frozen_time):
+    def test_request_max_allowed_time_w_refresh_timeout_no_error(self, frozen_time):
         tick_one_second = functools.partial(frozen_time.tick, delta=1.0)
 
         credentials = mock.Mock(
@@ -257,15 +275,17 @@
         )
 
         authed_session = google.auth.transport.requests.AuthorizedSession(
-            credentials, refresh_timeout=1.9
+            credentials, refresh_timeout=1.1
         )
         authed_session.mount(self.TEST_URL, adapter)
 
-        # The timeout is long, but the short refresh timeout will prevail.
-        with pytest.raises(requests.exceptions.Timeout):
-            authed_session.request("GET", self.TEST_URL, timeout=60)
+        # A short configured refresh timeout does not affect max_allowed_time.
+        # The latter is not adjusted to it and is only concerned with the actual
+        # execution time. The call below should thus not raise a timeout error
+        # (and `timeout` does not come into play either, as it's very long).
+        authed_session.request("GET", self.TEST_URL, timeout=60, max_allowed_time=3.1)
 
-    def test_request_timeout_w_refresh_timeout_and_tuple_timeout(self, frozen_time):
+    def test_request_timeout_w_refresh_timeout_timeout_error(self, frozen_time):
         tick_one_second = functools.partial(frozen_time.tick, delta=1.0)
 
         credentials = mock.Mock(
@@ -284,7 +304,10 @@
         )
         authed_session.mount(self.TEST_URL, adapter)
 
-        # The shortest timeout will prevail and cause a Timeout error, despite
-        # other timeouts being quite long.
+        # An UNAUTHORIZED response triggers a refresh (an extra request), thus
+        # the final request that otherwise succeeds results in a timeout error
+        # (all three requests together last 3 mocked seconds).
         with pytest.raises(requests.exceptions.Timeout):
-            authed_session.request("GET", self.TEST_URL, timeout=(100, 2.9))
+            authed_session.request(
+                "GET", self.TEST_URL, timeout=60, max_allowed_time=2.9
+            )