Moyes revealed Fellaini had an operation at the weekend after a troublesome back complaint failed to clear up as hoped.
"He had his operation on Saturday morning," said Moyes.
"We just couldn't get him recovered from his back. We thought it would be okay after 10 days but it wasn't getting any better so we made a quick decision to get the wrist done.
"We think it will be six weeks but there is a chance it could be shorter than that."
If Fellaini's recovery goes as scheduled, he will miss seven Premier League games, including the trip to Chelsea on January 19, plus both legs of the Capital One Cup semi-final with Sunderland and the FA Cup third and fourth rounds.
Crucially, though, the Belgium midfielder will be back well ahead of the trip to Olympiacos on February 25 in the first leg of United's Champions League knockout round tie.
Moyes confirmed the back injury should recover with rest and he anticipated Fellaini, who was back in Manchester on Monday, would go on to show the form which got him a summer move from Everton in the first place.
"His back problem is going to take time. That is a muscle injury which wasn't healing," said Moyes.
"It is unlucky. He has not felt 100 per cent because of his wrist. He has been finding it difficult.
"I asked him to see if he could get over the Christmas period. He got the back injury against Everton and since then he has never really been right.
"I have no doubt he will eventually be a good player for us."
Fellaini will miss the Boxing Day trip to Hull, when Robin van Persie (thigh) and Michael Carrick (Achilles) are also set to be missing.
However, Moyes is refusing to abandon all hope.
"This is the time when miracles happen," said Moyes.
"Let us hope there is one for Robin Van Persie and Michael Carrick."
Meanwhile, Moyes has confirmed he will make a decision after the festive fixtures about what course of action to take with the loans of Nick Powell, Jesse Lingard and Ryan Tunnicliffe.
The trio have all impressed to varying degrees at Wigan, Birmingham and Ipswich.
And, with new faces possibly proving difficult to get hold of during next month's transfer window, Moyes has three alternative options.
"We will try in January," he said.
"I am quite actively looking at our own boys. The likes of Jesse Lingard, Nick Powell and Ryan Tunnicliffe.
"They are the future. They are part of the club. They have been in the system.
"We hope they might be able to come back and help if we need it in January."