The Galatasaray coach believes his former club Manchester City, along with the likes of Chelsea, have overtaken United by bringing in new players to bolster their respective squads.
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"Chelsea changed and bought some new players, City bought four or five new players," Mancini told BBC World Service. "After many years, it's the time [for United] to refresh the team."
Moyes was widely praised for the job he did at Everton over the course of more than a decade, although he never won a trophy, and Mancini is confident that the three-time LMA Manager of the Year will get it right at Old Trafford.
"I think there is a good chance for him - he worked for 10 years at Everton and he did a really good job," the Italian added. "He is a good manager and I don't think he needs my advice."
One of the biggest challenges facing Moyes is the weight of his predecessor's legacy, with Sir Alex Ferguson turning United into England's most decorated club of the Premier League era.
"Alex was at the top but for every manager it is difficult [to come] after him," he added. "It's totally different to manage a team that should win every game, that every year [should] win a title or the Champions League."