diff --git a/pkg/kv/kvserver/replica_learner_test.go b/pkg/kv/kvserver/replica_learner_test.go
index 3f0227bc2937..d0e9f1ffef98 100644
--- a/pkg/kv/kvserver/replica_learner_test.go
+++ b/pkg/kv/kvserver/replica_learner_test.go
@@ -1367,6 +1367,9 @@ func TestReplicaGCQueueSeesLearnerOrJointConfig(t *testing.T) {
 func TestRaftSnapshotQueueSeesLearner(t *testing.T) {
 	defer leaktest.AfterTest(t)()
 	defer log.Scope(t).Close(t)
+
+	skip.UnderRace(t, "probable OOM")
+
 	ctx := context.Background()
 	blockSnapshotsCh := make(chan struct{})
 	knobs, ltk := makeReplicationTestKnobs()
diff --git a/pkg/sql/stats/stats_cache_test.go b/pkg/sql/stats/stats_cache_test.go
index cf2e52c0eda8..f66054319dcb 100644
--- a/pkg/sql/stats/stats_cache_test.go
+++ b/pkg/sql/stats/stats_cache_test.go
@@ -30,6 +30,7 @@ import (
 	"github.com/cockroachdb/cockroach/pkg/sql/types"
 	"github.com/cockroachdb/cockroach/pkg/testutils"
 	"github.com/cockroachdb/cockroach/pkg/testutils/serverutils"
+	"github.com/cockroachdb/cockroach/pkg/testutils/skip"
 	"github.com/cockroachdb/cockroach/pkg/testutils/sqlutils"
 	"github.com/cockroachdb/cockroach/pkg/util/encoding"
 	"github.com/cockroachdb/cockroach/pkg/util/leaktest"
@@ -435,6 +436,8 @@ func TestCacheAutoRefresh(t *testing.T) {
 	defer leaktest.AfterTest(t)()
 	defer log.Scope(t).Close(t)
 
+	skip.UnderRace(t, "probable OOM")
+
 	ctx := context.Background()
 	tc := serverutils.StartCluster(t, 3 /* numNodes */, base.TestClusterArgs{})
 	defer tc.Stopper().Stop(ctx)