diff --git a/x-pack/plugin/enrich/src/main/java/org/elasticsearch/xpack/enrich/action/TransportDeleteEnrichPolicyAction.java b/x-pack/plugin/enrich/src/main/java/org/elasticsearch/xpack/enrich/action/TransportDeleteEnrichPolicyAction.java
index c0dc07a3382ba..bf17ffaf6b956 100644
--- a/x-pack/plugin/enrich/src/main/java/org/elasticsearch/xpack/enrich/action/TransportDeleteEnrichPolicyAction.java
+++ b/x-pack/plugin/enrich/src/main/java/org/elasticsearch/xpack/enrich/action/TransportDeleteEnrichPolicyAction.java
@@ -9,6 +9,7 @@
 import org.elasticsearch.ResourceNotFoundException;
 import org.elasticsearch.action.ActionListener;
 import org.elasticsearch.action.admin.indices.delete.DeleteIndexRequest;
+import org.elasticsearch.action.admin.indices.get.GetIndexRequest;
 import org.elasticsearch.action.support.ActionFilters;
 import org.elasticsearch.action.support.IndicesOptions;
 import org.elasticsearch.action.support.master.AcknowledgedResponse;
@@ -126,7 +127,12 @@ protected void masterOperation(
             return;
         }
 
-        deleteIndicesAndPolicy(request.getName(), ActionListener.wrap((response) -> {
+        GetIndexRequest indices = new GetIndexRequest().indices(EnrichPolicy.getBaseName(request.getName()) + "-*")
+            .indicesOptions(IndicesOptions.lenientExpand());
+
+        String[] concreteIndices = indexNameExpressionResolver.concreteIndexNames(state, indices);
+
+        deleteIndicesAndPolicy(concreteIndices, request.getName(), ActionListener.wrap((response) -> {
             enrichPolicyLocks.releasePolicy(request.getName());
             listener.onResponse(response);
         }, (exc) -> {
@@ -135,10 +141,15 @@ protected void masterOperation(
         }));
     }
 
-    private void deleteIndicesAndPolicy(String name, ActionListener<AcknowledgedResponse> listener) {
-        // delete all enrich indices for this policy
-        DeleteIndexRequest deleteRequest = new DeleteIndexRequest().indices(EnrichPolicy.getBaseName(name) + "-*")
-            .indicesOptions(LENIENT_OPTIONS);
+    private void deleteIndicesAndPolicy(String[] indices, String name, ActionListener<AcknowledgedResponse> listener) {
+        if (indices.length == 0) {
+            deletePolicy(name, listener);
+            return;
+        }
+
+        // delete all enrich indices for this policy, we delete concrete indices here but not a wildcard index expression
+        // as the setting 'action.destructive_requires_name' may be set to true
+        DeleteIndexRequest deleteRequest = new DeleteIndexRequest().indices(indices).indicesOptions(LENIENT_OPTIONS);
 
         client.admin().indices().delete(deleteRequest, ActionListener.wrap((response) -> {
             if (response.isAcknowledged() == false) {
diff --git a/x-pack/plugin/enrich/src/test/java/org/elasticsearch/xpack/enrich/action/TransportDeleteEnrichPolicyActionTests.java b/x-pack/plugin/enrich/src/test/java/org/elasticsearch/xpack/enrich/action/TransportDeleteEnrichPolicyActionTests.java
index efdde89d3f310..911c41835d425 100644
--- a/x-pack/plugin/enrich/src/test/java/org/elasticsearch/xpack/enrich/action/TransportDeleteEnrichPolicyActionTests.java
+++ b/x-pack/plugin/enrich/src/test/java/org/elasticsearch/xpack/enrich/action/TransportDeleteEnrichPolicyActionTests.java
@@ -8,8 +8,11 @@
 
 import org.elasticsearch.ResourceNotFoundException;
 import org.elasticsearch.action.ActionListener;
+import org.elasticsearch.action.support.ActionTestUtils;
+import org.elasticsearch.action.support.DestructiveOperations;
 import org.elasticsearch.action.support.master.AcknowledgedResponse;
 import org.elasticsearch.cluster.service.ClusterService;
+import org.elasticsearch.common.settings.Settings;
 import org.elasticsearch.common.util.concurrent.EsRejectedExecutionException;
 import org.elasticsearch.common.xcontent.XContentType;
 import org.elasticsearch.index.IndexNotFoundException;
@@ -23,6 +26,7 @@
 import java.util.concurrent.CountDownLatch;
 import java.util.concurrent.atomic.AtomicReference;
 
+import static org.elasticsearch.test.hamcrest.ElasticsearchAssertions.assertAcked;
 import static org.elasticsearch.xpack.enrich.EnrichPolicyTests.randomEnrichPolicy;
 import static org.hamcrest.Matchers.equalTo;
 import static org.hamcrest.Matchers.nullValue;
@@ -115,6 +119,14 @@ public void testDeleteIsNotLocked() throws Exception {
         AtomicReference<Exception> error = saveEnrichPolicy(name, policy, clusterService);
         assertThat(error.get(), nullValue());
 
+        boolean destructiveRequiresName = randomBoolean();
+        if (destructiveRequiresName) {
+            Settings settings = Settings.builder()
+                .put(DestructiveOperations.REQUIRES_NAME_SETTING.getKey(), destructiveRequiresName)
+                .build();
+            assertAcked(client().admin().cluster().prepareUpdateSettings().setTransientSettings(settings));
+        }
+
         createIndex(EnrichPolicy.getBaseName(name) + "-foo1");
         createIndex(EnrichPolicy.getBaseName(name) + "-foo2");
 
@@ -151,6 +163,11 @@ public void onFailure(final Exception e) {
                 .get()
         );
 
+        if (destructiveRequiresName) {
+            Settings settings = Settings.builder().putNull(DestructiveOperations.REQUIRES_NAME_SETTING.getKey()).build();
+            assertAcked(client().admin().cluster().prepareUpdateSettings().setTransientSettings(settings));
+        }
+
         EnrichPolicyLocks enrichPolicyLocks = getInstanceFromNode(EnrichPolicyLocks.class);
         assertFalse(enrichPolicyLocks.captureExecutionState().isAnyPolicyInFlight());