The 30-year-old midfielder came off the bench on Sunday to secure a 2-1 stoppage-time win over Arsenal as Arsene Wenger's final trip to Old Trafford as Gunners manager ended in disappointment.
This could prove to be Fellaini's penultimate home appearance at the club he joined in 2013, given his contract is up at the end of the season and there has been an impasse in negotiations for much of the campaign.
However, it now looks like Mourinho has finally got his wish to get the Belgium international tied down to a new deal.
"I think Fellaini will stay," the United boss told BBC Radio 5 live. "We are trying to organise our team and are thinking about improvements.
"Until now, I haven't lost one second thinking about replacing him because I think he will stay."
Asked to expand on his comments in the post-match press conference, Mourinho added: "Yeah (I have spoken to Fellaini about his future).
"The position is that we are almost there - but in football almost is not enough.
"And 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 Ed Woodward's signature and Marouane's signature on it."