United boss David Moyes said on Friday the operation was a possibility due to Fellaini's separate back complaint.
Press Association Sport understands the Belgium star has already had the operation, which could rule him out until the end of February.
Fellaini has been struggling with the wrist injury for some time, although it had been possible for him to play through the discomfort.
Moyes always knew an operation would eventually be necessary, with the player preferring to go under the knife before the World Cup.
However, the separate back injury - which has forced him to miss five games since December 4's outing against his former club Everton and shows no immediate sign of clearing up - has led Moyes to get the wrist sorted.
It will now be hoped Fellaini will be back for the first leg of the Champions League last 16 encounter with Olympiacos on February 25.
Fellaini has struggled to make an impact since his £27.5million move from Goodison Park on transfer deadline day.
He has so far failed to score for his new club in a series of ineffective performances and on Friday, Moyes highlighted the current injury woe.
"Fellaini has an injury to his back, which we hoped might have healed up but hasn't,'' said the Scot.
"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.''