The Belgium international can depart Old Trafford on a free transfer at the end of the season, with his contract due to expire.
Per Cutts, a United source said:
"There is a fear that Marouane will only stay here with a major pay increase.
"He is on good money already, but feels he has a worth. They are in a difficult spot with him out of contract next summer.
"The club already took out their 12-month option to extend his deal back in January, so it's proving difficult.
"He wants to stay by all accounts, but has seen the club hand big money in recent seasons to new arrivals."
Former United boss David Moyes brought the 29-year-old to the club for £27.5 million, per Cutts, but he has remained a fringe player during his time in Manchester.
However, the big man has found a fan in current manager Jose Mourinho and has been given a starring role in the absence of Paul Pogba.
Fellaini has also won over the United faithful in the past year, with his name being sung by the Stretford End on a regular basis.
The player has already scored four goals this term, according to WhoScored.com, doubling his haul in the Premier League and Europe from last season.
In other United news, 18-year-old prodigy Krepin Diatta's representatives have cooled links with the Red Devils.
The player signed a new deal with Sarpsborg last February, and he is now focused on his career in Norway.
Speaking to FootMercato (h/t Tom Olver of Metro), Diatta's agent said: "The objective, for now, is to play. I don't think he's ready for a big European club. You need to go through the steps methodically."
Diatta recently told TV2 (h/t Olver) it would be a "dream" to transfer to Old Trafford if he was afforded the chance to switch to the Premier League.
Here is the teenager in action:
Mourinho has been complimentary about Fellaini's use since he took the reins at United 12 months ago, and it appears unlikely the manager will allow the midfielder to depart.
The Belgian gives United coverage as both a defensive and attacking option from the bench, and he remains one of the team's most dangerous players in the air and at set pieces.
United have not missed the influence of Pogba since losing the Frenchman to injury because of Fellaini's impact in his starting role.