diff --git a/setup.py b/setup.py
index 739b5ba00..60c0a144c 100644
--- a/setup.py
+++ b/setup.py
@@ -30,10 +30,7 @@
 release_status = "Development Status :: 5 - Production/Stable"
 dependencies = [
     "grpcio >= 1.38.1, < 2.0dev",  # https://github.com/googleapis/python-pubsub/issues/414
-    # NOTE: Maintainers, please do not require google-api-core>=2.x.x
-    # Until this issue is closed
-    # https://github.com/googleapis/google-cloud-python/issues/10566
-    "google-api-core[grpc] >= 1.31.5, <3.0.0dev,!=2.0.*,!=2.1.*,!=2.2.*,!=2.3.0",
+    "google-api-core[grpc] >= 2.8.0, <3.0.0dev",
     "proto-plus >= 1.15.0, <2.0.0dev",
     "protobuf >= 3.19.0, <4.0.0dev",
     "grpc-google-iam-v1 >=0.12.4, <1.0.0dev",
diff --git a/testing/constraints-3.6.txt b/testing/constraints-3.6.txt
deleted file mode 100644
index 20da08a34..000000000
--- a/testing/constraints-3.6.txt
+++ /dev/null
@@ -1,12 +0,0 @@
-# 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
-grpcio==1.38.1
-google-api-core==1.31.5
-libcst==0.3.10
-proto-plus==1.15.0
-grpc-google-iam-v1==0.12.4
-protobuf==3.19.0
diff --git a/testing/constraints-3.7.txt b/testing/constraints-3.7.txt
index 20da08a34..90fcc601c 100644
--- a/testing/constraints-3.7.txt
+++ b/testing/constraints-3.7.txt
@@ -5,7 +5,7 @@
 # e.g., if setup.py has "foo >= 1.14.0, < 2.0.0dev",
 # Then this file should have foo==1.14.0
 grpcio==1.38.1
-google-api-core==1.31.5
+google-api-core==2.8.0
 libcst==0.3.10
 proto-plus==1.15.0
 grpc-google-iam-v1==0.12.4