It is understood United will pay Monaco an initial £36m for the 19-year-old forward, making him the most expensive teenager in football history.
Martial was given permission to leave the France national squad on Monday but is now back with his compatriots ahead of their upcoming friendlies against Portugal and Serbia.
United are yet to confirm the transfer, but an announcement is expected later on deadline day.
Martial signed a new deal with Monaco only in June, when the Ligue 1 club’s vice-president Vadim Vasilyev called the player “untransferable”.
Vasilyev said then: “He was being contacted by some of Europe’s biggest clubs but we wanted to make sure that he continues his development with AS Monaco.
“What Anthony has been able to achieve at just 19 years of age is impressive, to say the least, but the best is yet to come.”
Martial’s imminent arrival comes after United sold Javier Hernandez to Bayer Leverkusen, while Adnan Januzaj is has secured a loan move to Borussia Dortmund.
Robin van Persie and Radamel Falcao have also left Old Trafford this summer, leaving Wayne Rooney as the club’s only recognised senior striker.