diff --git a/setup.py b/setup.py
index 719f885..ee7507e 100644
--- a/setup.py
+++ b/setup.py
@@ -26,7 +26,7 @@
 release_status = "Development Status :: 5 - Production/Stable"
 url = "https://github.com/googleapis/python-managed-identities"
 dependencies = [
-    "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",
 ]
diff --git a/testing/constraints-3.7.txt b/testing/constraints-3.7.txt
index f74740e..b450739 100644
--- a/testing/constraints-3.7.txt
+++ b/testing/constraints-3.7.txt
@@ -19,6 +19,6 @@
 #
 # e.g., if setup.py has "foo >= 1.14.0, < 2.0.0dev",
 # Then this file should have foo==1.14.0
-google-api-core==2.8.0
+google-api-core==1.32.0
 proto-plus==1.15.0
 protobuf==3.19.0