diff --git a/setup.py b/setup.py
index b5cc803..8031f5a 100644
--- a/setup.py
+++ b/setup.py
@@ -29,7 +29,7 @@
     # 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] >= 1.27.0, <3.0.0dev",
+    "google-api-core[grpc] >= 1.31.5, <3.0.0dev,!=2.0.*,!=2.1.*,!=2.2.*,!=2.3.0",
     "proto-plus >= 1.15.0",
     "packaging >= 14.3",
 ]
diff --git a/testing/constraints-3.6.txt b/testing/constraints-3.6.txt
index 1d6a015..e01269f 100644
--- a/testing/constraints-3.6.txt
+++ b/testing/constraints-3.6.txt
@@ -4,7 +4,6 @@
 # Pin the version to the lower bound.
 # e.g., if setup.py has "google-cloud-foo >= 1.14.0, < 2.0.0dev",
 # Then this file should have google-cloud-foo==1.14.0
-google-api-core==1.27.0
+google-api-core==1.31.5
 proto-plus==1.15.0
 packaging==14.3
-google-auth==1.24.0  # TODO: remove when google-auth>=1.25.0 is transitively required through google-api-core