ESPN says Mourinho wants United to agree terms with Fellaini over a new deal before the end of the year.
Fellaini is yet to agree to an extension with his current deal at Old Trafford running out at the end of the season.
It leaves United in a precarious position with Fellaini able to sign a pre-contract with another club in the new year.
But Mourinho is personally pushing for a new deal to be agreed for the Belgium international to ensure ward off interest in him.
Galatasaray are showing serious interest in Fellaini as new contract talks stall.