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The Belgium international has not featured since the 1-0 home defeat against his former club Everton early last month after undergoing surgery on his wrist.
Fellaini has also been troubled by a back problem, but Moyes is hopeful of having the summer signing available ahead of meetings with Sunderland in the Capital One Cup and Cardiff City in the Premier League.
Combative midfielder Fellaini will have his cast removed Monday and will then be looking to stake his claim for a place in the United starting lineup.
"Hopefully next week we'll get him back," Moyes told MUTV.
Fellaini has not reproduced the sort of form that he showed for Everton at the start of his United career, and will be eager to make an impact when he returns to the first team.