The 30-year-old is due to become a free agent in a little over a month's time and is reportedly a target for as many as six clubs, including Paris Saint-Germain and Juventus.
Fellaini showed that he can still play an important role at Old Trafford with his match-winning header against Arsenal on Sunday afternoon, however, which he followed up by tapping the club crest in celebration.
Asked about the Belgian's uncertain future on the back of his side's 2-1 win over Arsenal, Mourinho told BBC Sport: "I think a player that grabs the crest after scoring is telling clearly that he wants to stay.
"I want him to stay, the club wants him to stay, I think he will stay, it is nearly there."
Fellaini's goal was the first 90th-minute winner United have scored against Arsenal in the Premier League era.