diff --git a/packages/google-cloud-python-speech/setup.py b/packages/google-cloud-python-speech/setup.py
index 5ff1b3bf9161..f738550a0f2a 100644
--- a/packages/google-cloud-python-speech/setup.py
+++ b/packages/google-cloud-python-speech/setup.py
@@ -32,8 +32,8 @@
     # 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] >= 1.28.0, <3.0.0dev",
-    "proto-plus >= 1.4.0",
+    "google-api-core[grpc] >= 1.31.5, <3.0.0dev,!=2.0.*,!=2.1.*,!=2.2.*,!=2.3.0",
+    "proto-plus >= 1.15.0",
 ]
 extras = {"libcst": "libcst >= 0.2.5"}
 
diff --git a/packages/google-cloud-python-speech/testing/constraints-3.6.txt b/packages/google-cloud-python-speech/testing/constraints-3.6.txt
index 96d36040bdfe..2bdebaf039b3 100644
--- a/packages/google-cloud-python-speech/testing/constraints-3.6.txt
+++ b/packages/google-cloud-python-speech/testing/constraints-3.6.txt
@@ -5,6 +5,6 @@
 #
 # 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==1.28.0
+google-api-core==1.31.5
 libcst==0.2.5
-proto-plus==1.4.0
+proto-plus==1.15.0