Sky Sports News understands the new deal will see the French international earn £250,000 a week.
Martial's existing deal was set to expire at the end of this season before United activated a year-long extension clause in November.
Talks over an extension have been ongoing for months, with Martial rejecting an offer from United last October.
Martial clashed with then manager Jose Mourinho last summer when he returned to France from the club's preseason tour of the United States for the birth of his child.
His agent publicly stated in June that Martial wanted to leave because United had not demonstrated they wanted make the 23-year-old "an important member of the team".
But Martial has emerged as a key player for United this season under both Mourinho and Ole Gunnar Solskjaer, appearing in 22 of United's 24 Premier League matches.
Last modified on Thursday, 31 January 2019