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.
Mourinho has been quoted as saying that the United squad is weaker when Fellaini is not in it, and reports now suggest he will do all he can to get the midfielder tied down to a long-term deal with the Reds.
Fellaini is yet to start a game this season but inspired United to victory as a substitute in the Champions League against Basel on Tuesday.
Fellaini is 30-years-old in November and despite United having a policy of only offering one-year extensions to players of that age, Fellaini is thought to want a longer contract as he enjoys working under Mourinho.
The report, however, adds that Fellaini wants more than the one-year extension now on the table.
But various reports have hinted towards Fellaini, whose contract expires at the end of the season, opting for a move to Turkey over the summer.