diff --git a/tornado/iostream.py b/tornado/iostream.py
index 53e81fff3..198833c2a 100644
--- a/tornado/iostream.py
+++ b/tornado/iostream.py
@@ -1366,7 +1366,7 @@ def _do_ssl_handshake(self) -> None:
                     peer = self.socket.getpeername()
                 except Exception:
                     peer = "(not connected)"
-                gen_log.warning(
+                gen_log.info(
                     "SSL Error on %s %s: %s", self.socket.fileno(), peer, err
                 )
                 return self.close(exc_info=err)
diff --git a/tornado/test/iostream_test.py b/tornado/test/iostream_test.py
index ca909f276..040228ea6 100644
--- a/tornado/test/iostream_test.py
+++ b/tornado/test/iostream_test.py
@@ -1066,7 +1066,7 @@ def test_start_tls_smtp(self):
     def test_handshake_fail(self):
         server_future = self.server_start_tls(_server_ssl_options())
         # Certificates are verified with the default configuration.
-        with ExpectLog(gen_log, "SSL Error"):
+        with ExpectLog(gen_log, "SSL Error", level=logging.INFO):
             client_future = self.client_start_tls(server_hostname="localhost")
             with self.assertRaises(ssl.SSLError):
                 yield client_future
@@ -1077,7 +1077,7 @@ def test_handshake_fail(self):
     def test_check_hostname(self):
         # Test that server_hostname parameter to start_tls is being used.
         server_future = self.server_start_tls(_server_ssl_options())
-        with ExpectLog(gen_log, "SSL Error"):
+        with ExpectLog(gen_log, "SSL Error", level=logging.INFO):
             client_future = self.client_start_tls(
                 ssl.create_default_context(), server_hostname="127.0.0.1"
             )
@@ -1231,7 +1231,7 @@ async def test_no_match(self):
         with ExpectLog(
             gen_log,
             ".*alert bad certificate",
-            level=logging.WARNING,
+            level=logging.INFO,
             required=platform.system() != "Windows",
         ):
             with self.assertRaises(ssl.SSLCertVerificationError):