According to ESPN FC's Rob Dawson, sources within the club stated negotiations will resume after a first round of talks in December did not result in an agreement.
Martial's current deal will run out next summer, with an option for an additional year included.
Per the report, the 22-year-old was one of the players Arsenal enquired about in the negotiations for Alexis Sanchez in January. United ended up parting with Henrikh Mkhitaryan in that deal.
Martial finds himself in a tricky spot at Old Trafford. The young Frenchman has shown plenty of flashes of his tremendous talent, but he has failed to put together a consistent stretch of good performances since Jose Mourinho became manager.
A lot of his minutes have come in a wide role or from the bench, and the arrival of Sanchez can only hurt his playing time moving forward. Per Dawson, the former AS Monaco man will "seek assurances" about his role, and that could be key to a breakthrough.
There's little doubt about his value, as he has scored nine goals and added five assists in the Premier League this season.
The additional year in his current contract gives the Red Devils plenty of time to work out a deal that suits all parties, so the club should feel optimistic as negotiations resume.