The 50-year-old Scot, who replaced Sir Alex Ferguson this summer, has only managed to secure three wins from his opening six seven top-flight games in charge of the Red Devils.
Moyes gave Coleman a chance to impress during his distinguished reign at Everton before moving to Old Trafford, and the Republic of Ireland defender is confident United will improve under his old boss.
“Just from my time with him, all I can say is he has been a top manager and he has done so well with Everton,” Coleman said.
“He got the job deservedly – he was recommended by Sir Alex himself – so I think he will definitely turn it around.
“It’s only early days, only a few games into the season, so I am sure he will.”
Coleman added he is satisfied with Everton’s form in recent weeks under new manager Roberto Martínez despite a disappointing 3-1 loss at Manchester City on Saturday.
“It’s been going quite well. He has come in and he has put his own philosophy on the team, and we have done quite well to adapt to it.
“We have got very good players. The [Manchester] City game [a 3-1 defeat on Saturday] was a disappointment, but hopefully we can put that right with the next game at home to Hull.
“But it’s been a good start and we have all enjoyed it. It’s different. You can see that he likes to play out from the back a bit more, but he has been great for us and hopefully we can have a good, successful season under him.”
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