Van Persie suffered the injury during the 1-0 Champions League win over Shakhtar Donetsk on Tuesday.
"We've not had them for quite a period recently."
The Netherlands striker, 30, has scored seven goals in 11 Premier League matches this season and his absence comes as a huge blow to struggling boss Moyes, whose side lie ninth in the Premier League after a poor start to the campaign.
The biggest blow could come if the rehabiliation takes any longer than originally expected: United face a crunch tie at Chelsea on January 19, and Moyes will be desperately hoping that his top striker is fit in time for that game.
After a successful but initially injury-prone spell at Arsenal, Van Persie appeared to have found full fitness before joining United in the summer of 2012.
He went on to score 26 Premier League goals after featuring in every match last season, helping United win the title.
Which matches will Van Persie miss?
15 December: Aston Villa v Manchester United - Premier League
21 December: Manchester United v West Ham United - Premier League
26 December: Hull City v Manchester United - Premier League
28 December: Norwich City v Manchester United - Premier League
1 January: Manchester United v Tottenham Hotspur - Premier League
5 January: Manchester United v Swansea City - FA Cup
11 January: Manchester United v Swansea City - Premier League
19 January: Chelsea v Manchester United - Premier League
28 January: Manchester United v Cardiff City - Premier League