The Portugal international has not featured for United since the 1-0 defeat to Newcastle at Old Trafford on 7 December.
Nani has recently been linked with a move to Juventus in the January transfer window, but the winger will have to overcome his injury first.
Moyes said: "Nani has a hamstring injury which is a little bit worse than we thought. He is probably going to be missing for a few weeks."
United are already without the services of Robin van Persie until January, after the striker suffered a thigh strain.
And Wayne Rooney is struggling with a groin injury as United prepare for Saturday's visit of West Ham in the Premier League.