The Belgium international midfielder only has until the end of the season to run on his current deal, prompting speculation about a possible exit in the summer.
It appears as though Fellaini has been offered fresh terms, though, and Mourinho has joked that the 30-year-old should make a quick decision, per Rob Dawson of ESPN FC.
"I told him yesterday, sign the contract and shut up," he said. "I told him, don't gamble too much. Sign the contract before you know the extent of the injury."
Fellaini hobbled out of United's loss to Tottenham Hotspur in midweek and is set for a "few weeks" on the sidelines, according to Mourinho.
Dawson added that Fellaini was offered a new deal by the Red Devils in September but declined to sign as he wants a longer contract than is being offered. His position means he can potentially agree a pre-contract with a club overseas with a view to a free transfer in the summer, with interest from Italy and Turkey.
As noted by Kristof Terreur of Het Laatste Nieuws, it's been a frustrating season for Fellaini, who has suffered three significant injury layoffs:
Kristof Terreur 📰 @HLNinEngeland
Marouane Fellaini limping through mixed-zone yesterday. Confirms that he’s injured: “It’s the knee again.” He injured his knee on October 7th during 🇧🇦-🇧🇪. Since then: 🚑 out for 4 weeks ✅ 4 games (148 mins) 🚑 out for 6 weeks ✅ 3 games (35 mins) 🚑 out for ? weeks #MUFC https://t.co/NTdAPHptEo
Per Dawson, Mourinho has been full of praise for the often maligned Belgian in the past and has made it clear he would like him to stay.
During his time at United, the rangy midfielder has been ridiculed by some supporters, as his playing style isn't often the most easy on the eye. Fellaini sometimes slows the pace in midfield, while his powerful physique means he isn't the quickest when carrying the ball.
Additionally, there have been times when his hot head has cost United, as Fellaini can make rash decisions and has a tendency for using his elbows a little too much in aerial battles.
Football writer Liam Canning also doesn't believe the club should bow to any excessive demands being made by the player as part of contract negotiations:
Still, when utilised effectively, Fellaini can be a massive threat. The Belgian can dominate defences with his aerial prowess and ability to take the ball down on his chest. Fellaini has also chipped in with some useful goals for the Red Devils, netting in three semi-finals for the team down the years.
And as noted by ESPN FC, Fellaini was in prolific form when he was playing regularly earlier in the season:
Even so, the midfielder is not someone United will be seeking to build their team around in the future, especially given Mourinho can call upon the likes of Paul Pogba, Nemanja Matic and Ander Herrera in central areas.
From the player's perspective, taking some time to consider his options is understandable. Although United are an enormous club managed by a coach that has total faith in Fellaini, the chance to go abroad and play for a team where he'd have a much more important role would be tempting.