The report claims Fellaini has decided a reunion with former-Toffees boss David Moyes would be his first choice this summer, and he's keen for the Premier League champions to make a bid for him.
The Belgian international's release clause is reported to be worth £23.5m, but neither United nor Arsenal have matched it yet.
United have made Barcelona midfielder Cesc Fabregas their top summer priority having missed out on his old teammate Thiago Alcantara.
The Old Trafford club reportedly lodged a £26m bid for the former-Arsenal captain - but that was rejected.
Moyes is expected to go back in for the Spain international in the next few days with an offer around £30m.
Arsenal are still trying to wrap up a deal for a big-name striker, with Gonzalo Higuain heavily linked with the club.
Both clubs are thought to be interested in the Everton midfielder, but they'll have to move fast to beat his release clause deadline.
The £23.5m buy-out clause expires on August 14, according to reports, after which Everton can demand as much as they think he's worth - if they want to sell at all.
And the 25-year-old would reportedly prefer to remain in the north west, where he's settled, suggesting a switch to United would be preferred to north London.