The Sun reports that Mourinho wants to reward the Argentine defender for his consistency his season.
Rojo had struggled for regular game time under Louis van Gaal but has notched 34 appearances under Mourinho this term.
The 27-year-old is under contract for a further two years but could put pen to a new 12-month deal that will see him remain at Old Trafford until at least 2020, earning £100,000 per week.
A United source said: “Jose has been impressed by his desire and attitude and is happy for him to stay.
“They will all sit down at the end of the season and discuss it. Rojo also has no issues with remaining.
“He does have two years left on his current money, but that will stay the same and they'll be no increase.
“It is just to keep things comfortable for everyone. Jose likes Marcos and things are going well for them both."