The 29-year-old is yet to start a game for United this season, but the midfielder is a firm favourite of head coach Jose Mourinho, who has been keen to stress Fellaini's importance to the team over the last couple of weeks.
It is understood that Galatasaray tried to sign the former Everton number 10 during the summer, but Mourinho rejected the interest in a player that is considered a key part of the Portuguese's squad.
Fellaini's current deal will expire at the end of the season, but according to The Mirror, United have tabled an initial offer to the midfielder, who could start Sunday's Premier League game against Everton at Old Trafford.
The report, however, adds that Fellaini wants more than the one-year extension currently on the table.
The midfielder has scored twice in five appearances for United during the 2017-18 campaign - including one in the team's 3-0 victory over Basel in the Champions League on Tuesday night.