diff --git a/setup.py b/setup.py
index 507a5fb84..de49ce3d6 100644
--- a/setup.py
+++ b/setup.py
@@ -30,7 +30,7 @@
 release_status = "Development Status :: 5 - Production/Stable"
 dependencies = [
     "grpcio >= 1.38.1, < 2.0dev",  # https://github.com/googleapis/python-pubsub/issues/414
-    "google-api-core[grpc] >= 2.8.0, <3.0.0dev",
+    "google-api-core[grpc] >= 1.32.0, <3.0.0dev,!=2.0.*,!=2.1.*,!=2.2.*,!=2.3.*,!=2.4.*,!=2.5.*,!=2.6.*,!=2.7.*",
     "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-3.7.txt b/testing/constraints-3.7.txt
index 90fcc601c..38e0f719d 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==2.8.0
+google-api-core==1.32.0
 libcst==0.3.10
 proto-plus==1.15.0
 grpc-google-iam-v1==0.12.4