diff --git a/sentry_sdk/client.py b/sentry_sdk/client.py
index 2c245297bd..928fc3ea8b 100644
--- a/sentry_sdk/client.py
+++ b/sentry_sdk/client.py
@@ -1006,6 +1006,13 @@ def _capture_metric(self, metric):
         metric["attributes"]["sentry.sdk.name"] = SDK_INFO["name"]
         metric["attributes"]["sentry.sdk.version"] = SDK_INFO["version"]
 
+        server_name = self.options.get("server_name")
+        if (
+            server_name is not None
+            and SPANDATA.SERVER_ADDRESS not in metric["attributes"]
+        ):
+            metric["attributes"][SPANDATA.SERVER_ADDRESS] = server_name
+
         environment = self.options.get("environment")
         if environment is not None and "sentry.environment" not in metric["attributes"]:
             metric["attributes"]["sentry.environment"] = environment
diff --git a/tests/test_metrics.py b/tests/test_metrics.py
index c7b786beb4..0a1736a537 100644
--- a/tests/test_metrics.py
+++ b/tests/test_metrics.py
@@ -7,6 +7,7 @@
 from sentry_sdk import get_client
 from sentry_sdk.envelope import Envelope
 from sentry_sdk.types import Metric
+from sentry_sdk.consts import SPANDATA, VERSION
 
 
 def envelopes_to_metrics(envelopes):
@@ -93,7 +94,7 @@ def test_metrics_experimental_option(sentry_init, capture_envelopes):
 
 
 def test_metrics_with_attributes(sentry_init, capture_envelopes):
-    sentry_init(release="1.0.0", environment="test")
+    sentry_init(release="1.0.0", environment="test", server_name="test-server")
     envelopes = capture_envelopes()
 
     sentry_sdk.metrics.count(
@@ -110,6 +111,10 @@ def test_metrics_with_attributes(sentry_init, capture_envelopes):
     assert metrics[0]["attributes"]["sentry.release"] == "1.0.0"
     assert metrics[0]["attributes"]["sentry.environment"] == "test"
 
+    assert metrics[0]["attributes"][SPANDATA.SERVER_ADDRESS] == "test-server"
+    assert metrics[0]["attributes"]["sentry.sdk.name"].startswith("sentry.python")
+    assert metrics[0]["attributes"]["sentry.sdk.version"] == VERSION
+
 
 def test_metrics_with_user(sentry_init, capture_envelopes):
     sentry_init()