diff --git a/test_runner/regress/test_import_pgdata.py b/test_runner/regress/test_import_pgdata.py
index ba60c3caa6..8d4f908cc0 100644
--- a/test_runner/regress/test_import_pgdata.py
+++ b/test_runner/regress/test_import_pgdata.py
@@ -287,6 +287,17 @@ def test_pgdata_import_smoke(
     with pytest.raises(psycopg2.errors.UndefinedTable):
         br_initdb_endpoint.safe_psql(f"select * from {workload.table}")
 
+    # The storage controller might be overly eager and attempt to finalize
+    # the import before the task got a chance to exit.
+    env.storage_controller.allowed_errors.extend(
+        [
+            ".*Call to node.*management API.*failed.*Import task still running.*",
+        ]
+    )
+
+    for ps in env.pageservers:
+        ps.allowed_errors.extend([".*Error processing HTTP request.*Import task not done yet.*"])
+
 
 @run_only_on_default_postgres(reason="PG version is irrelevant here")
 def test_import_completion_on_restart(
@@ -471,6 +482,17 @@ def test_import_respects_timeline_lifecycle(
     else:
         raise RuntimeError(f"{action} param not recognized")
 
+    # The storage controller might be overly eager and attempt to finalize
+    # the import before the task got a chance to exit.
+    env.storage_controller.allowed_errors.extend(
+        [
+            ".*Call to node.*management API.*failed.*Import task still running.*",
+        ]
+    )
+
+    for ps in env.pageservers:
+        ps.allowed_errors.extend([".*Error processing HTTP request.*Import task not done yet.*"])
+
 
 @skip_in_debug_build("Validation query takes too long in debug builds")
 def test_import_chaos(