diff --git a/distributed/client.py b/distributed/client.py
index bbda46acf4..5ce872c064 100644
--- a/distributed/client.py
+++ b/distributed/client.py
@@ -2581,7 +2581,7 @@ def _graph_to_futures(
             if not isinstance(dsk, HighLevelGraph):
                 dsk = HighLevelGraph.from_collections(id(dsk), dsk, dependencies=())
 
-            dsk = highlevelgraph_pack(dsk, keyset, self, self.futures)
+            dsk = highlevelgraph_pack(dsk, self, keyset)
 
             if isinstance(retries, Number) and retries > 0:
                 retries = {k: retries for k in dsk}
diff --git a/distributed/protocol/highlevelgraph.py b/distributed/protocol/highlevelgraph.py
index 77de669ebb..512a7abd4b 100644
--- a/distributed/protocol/highlevelgraph.py
+++ b/distributed/protocol/highlevelgraph.py
@@ -27,9 +27,8 @@ def _materialized_layer_pack(
     layer: Layer,
     all_keys,
     known_key_dependencies,
+    client,
     client_keys,
-    allowed_client,
-    allowed_futures,
 ):
     from ..client import Future
 
@@ -47,11 +46,11 @@ def _materialized_layer_pack(
     dsk = {k: unpack_remotedata(v, byte_keys=True) for k, v in layer.items()}
     unpacked_futures = set.union(*[v[1] for v in dsk.values()]) if dsk else set()
     for future in unpacked_futures:
-        if future.client is not allowed_client:
+        if future.client is not client:
             raise ValueError(
                 "Inputs contain futures that were created by another client."
             )
-        if tokey(future.key) not in allowed_futures:
+        if tokey(future.key) not in client.futures:
             raise CancelledError(tokey(future.key))
     unpacked_futures_deps = {}
     for k, v in dsk.items():
@@ -76,15 +75,13 @@ def _materialized_layer_pack(
     return {"dsk": dsk, "dependencies": dependencies}
 
 
-def highlevelgraph_pack(
-    hlg: HighLevelGraph, client_keys, allowed_client, allowed_futures
-):
+def highlevelgraph_pack(hlg: HighLevelGraph, client, client_keys):
     layers = []
 
     # Dump each layer (in topological order)
     for layer in (hlg.layers[name] for name in hlg._toposort_layers()):
         if not layer.is_materialized():
-            state = layer.__dask_distributed_pack__()
+            state = layer.__dask_distributed_pack__(client)
             if state is not None:
                 layers.append(
                     {
@@ -104,9 +101,8 @@ def highlevelgraph_pack(
                     layer,
                     hlg.get_all_external_keys(),
                     hlg.key_dependencies,
+                    client,
                     client_keys,
-                    allowed_client,
-                    allowed_futures,
                 ),
             }
         )