The Dutchman parted company with the Premier League champions earlier this week, but is hoping that the team continue their winning tradition under the ex-Everton boss.
"Let me put one thing straight, I hope United are going to experience as much success under David Moyes as I've had," he told Sky Sports News.
"I hope [United win the league]. I hope for United and for the fantastic fans. United is a club that has seen many, many successes over the years and know how to win.
"I'm sure David Moyes and the staff will be clever enough to use the experience and I'm sure with a couple of additions United will be very strong contenders to win the Premier League."
Meulensteen has been linked with a coaching role at Russian club Anzhi Makhachkala.