French outlet RMC (h/t Metro) reported "tensions have thawed" between Martial and Mourinho, while executive-vice chairman Ed Woodward also remains a fan of the Frenchman.
The report also stated the new deal could be worth as much as £175,000 per week. The Daily Star believes he's currently on £65,000 per week, but that's not confirmed.
The 22-year-old has been a regular in the starting XI in recent weeks after starting the season as a depth option. He has only scored three Premier League goals so far, but his play has been encouraging and his uptick in form has coincided with Alexis Sanchez's struggles.
Martial has recently talked up his relationship with the coach:
The youngster's current deal will run out in 2019, but the club has an option to extend it for an additional year. The uncertainty surrounding his status led to transfer rumours in the summer, and just last week RMC Sport (h/t Calciomercato.com) reported Martial had turned down several renewal offers amid interest from Juventus.
A transfer seemed unlikely, however, as the Red Devils spent big on the former Monaco man just a few years ago and his development has stalled since. Interested clubs likely wouldn't be willing to come close to that valuation at this point.
United signed Luke Shaw to a new contract earlier this month, and many assume Martial is next:
It hardly seems a coincidence these rumours have emerged just days after Martial bagged a brace against Chelsea, taking his tally to three goals in his last two matches. That gives him some leverage entering renewed talks with the club, and the time to push for an improved contract seems ideal.