The Dutchman, who left the Premier League champions on Wednesday, believes the Scot can maintain the club's winning tradition but was unsure whether Wayne Rooney will stay
The Dutchman's departure from the club was confirmed earlier on Wednesday after 12 years - over two spells - at the Premier League champions.
And with United set to start a new era under Moyes, Meulensteen wished the former Everton manager well, insisting he can maintain the club's winning mentality.
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"Let me put one thing straight, I hope United are going to experience as much success under David Moyes as I’ve had.
"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."
One of Moyes' first challenges at Old Trafford is to resolve Wayne Rooney's future, with Goal exclusively revealing the striker has held secret talks with two clubs over a summer move.
Meulensteen is unsure if the 27-year-old will stay, but is keen for him to do so.
"Wayne has his own agenda, he’s a young player still, in his prime," he added. "I hope he stays but its obviously not up to me to decide and whatever he decides to do I wish him all the best."
Meulensteen briefly left his role at United for an ill-fated short-lived spell at Brondby, but is hoping to branch out on his own again.
Russian side Anzhi Makhachkala have been linked as a possible destination for a coaching role, but the Dutchman was coy on a possible move.
"I've been speaking to various people," he said. "Obviously I know Guus Hiddink, he’s a fellow countryman, let’s see what the future brings.
"I know whenever I need Sir Alex Ferguson he’ll only be a phone call away."