Both Premiership giants have been linked with the Bosman prospect.
But Desailly told fourfourtwo.com: "A few years ago I would have thought he would go to Arsenal, but now that he is 28 he needs to go to a team that - with him or without him - will succeed, in order to put pressure on him to deliver performances.
"At Arsenal there are younger players and you need to talk to them, surround them and have a positive attitude around them.
"In Joe's current situation he needs to push on. He doesn't need to help the others, he needs a club to help him discover his best levels once again. Therefore, I would say Manchester United."