Fellaini headed home a stoppage-time winner at Old Trafford, giving Mourinho's men a 2-1 win over the Gunners on Sunday.
Fellaini's contract is set to expire at the end of the season and the 30-year-old has yet to sign fresh terms, leading to speculation he could depart on a free transfer.
But, after the Belgian's dramatic late goal, the Man Utd boss said he expects a new deal for Fellaini imminently.
"I think a player that grabs the crest after scoring is telling clearly that he wants to stay," Mourinho told the BBC .
"I want him to stay, the club wants him to stay. I think he will stay, it is nearly there."
Speaking in the post-match press conference, Mourinho added: “I believe he will stay. He’s a good player and he can be used in many different circumstances, according to the nature of the game.
"That’s why we want him to stay and that’s why I believe that he is going to stay.
"The position is we are almost there. But, in football, almost is not enough. I think we are almost there but I can't celebrate because almost is not enough.
"I want to see the white paper with the United crest and Mr Ed Woodward's signature and Maraoune's signature on it."
Fellaini has made 22 appearances in all completions for United in 2017-18, scoring five goals.