United have been linked with a move for Griezmann for quite some time and are expected to make a serious move for him this summer.
However, Hargreaves thinks that United should pursue Dybala instead. The Argentine has scored 17 goals and provided eight assists for Juventus this season.
He told BT Sport: "You have to look at what you have and what you desperately need, I think they need a Dybala-type player.
"Everybody speaks about Griezmann, I think Griezmann would be terrific. But I think Dybala, he's a little bit younger, he's going to be a superstar if he isn't already.
"He would be magnificent. That star power that Ibra brought, I think Dybala could bring that as well."