Fellaini has not featured since United lost to his former club Everton at Old Trafford on 4 December, after suffering a problem with his back.
Moyes had expected the Belgium international to be fit for the festive schedule, but Fellaini has already been ruled out of Saturday's Premier League match at home to West Ham.
The Scot is seeking further medical advice on the matter, and if the outcome is not positive, it is likely Fellaini will have the wrist operation he has been putting off for some time.
"Fellaini has an injury to his back, which we hoped might have healed up but it hasn't," Moyes said.
"I am going to get another scan and if it hasn't healed up then I might need to take the decision to get his wrist done as well.
"We hoped his back would be better in 10 days but he is still feeling it."
Moyes did receive better news about Michael Carrick, who has not featured since the victory over Arsenal at the beginning of last month.
The midfielder should be available to face former club Tottenham on New Year's Day after recovering from his Achilles injury.
"Michael has just started some work outside on the grass," said Moyes. "He is getting a little bit closer.
"He is roughly due to be back at New Year's Day time. We might be able to squeeze a few more days out of it but I couldn't be sure yet."