diff --git a/setup.py b/setup.py
index 1bb092f..4e5b94c 100644
--- a/setup.py
+++ b/setup.py
@@ -32,7 +32,7 @@
     # NOTE: Maintainers, please do not require google-cloud-core>=2.x.x
     # Until this issue is closed
     # https://github.com/googleapis/google-cloud-python/issues/10566
-    "google-cloud-core >= 1.3.0, < 3.0dev",
+    "google-cloud-core >= 1.4.4, < 3.0dev",
 ]
 extras = {}
 
diff --git a/testing/constraints-2.7.txt b/testing/constraints-2.7.txt
deleted file mode 100644
index e69de29..0000000
diff --git a/testing/constraints-3.6.txt b/testing/constraints-3.6.txt
deleted file mode 100644
index dde085b..0000000
--- a/testing/constraints-3.6.txt
+++ /dev/null
@@ -1,22 +0,0 @@
-# Copyright 2021 Google LLC
-#
-# Licensed under the Apache License, Version 2.0 (the "License");
-# you may not use this file except in compliance with the License.
-# You may obtain a copy of the License at
-#
-# http://www.apache.org/licenses/LICENSE-2.0
-#
-# Unless required by applicable law or agreed to in writing, software
-# distributed under the License is distributed on an "AS IS" BASIS,
-# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
-# See the License for the specific language governing permissions and
-# limitations under the License.
-#
-# This constraints file is used to check that lower bounds
-# are correct in setup.py
-# List *all* library dependencies and extras in this file.
-# Pin the version to the lower bound.
-#
-# e.g., if setup.py has "foo >= 1.14.0, < 2.0.0dev",
-# Then this file should have foo==1.14.0
-google-cloud-core==1.3.0
diff --git a/testing/constraints-3.7.txt b/testing/constraints-3.7.txt
index dde085b..e37a6a7 100644
--- a/testing/constraints-3.7.txt
+++ b/testing/constraints-3.7.txt
@@ -19,4 +19,4 @@
 #
 # e.g., if setup.py has "foo >= 1.14.0, < 2.0.0dev",
 # Then this file should have foo==1.14.0
-google-cloud-core==1.3.0
+google-cloud-core==1.4.4