The striker, who has been linked with Arsenal, was allowed to leave the France squad on Monday for a medical with Louis van Gaal's side.
Should he sign, Martial, who scored eight goals in 31 Ligue 1 games last season. will become the costliest teenager in football history.
He will also become Manchester United's third most expensive signing.
The Red Devils paid £59.7m to Real Madrid for Argentina winger Angel Di Maria and £37.1m to Chelsea for Spain midfielder Juan Mata.
United have until 18:00 BST on Tuesday, when the transfer window shuts for Premier League clubs, to complete the deal.
A France Under-21 international who has just been called up to the senior national team for the first time.
The teenager scored 32 goals in just 21 games at youth level for Lyon before breaking into the first team in the 2012-13 season.
He made just three appearances before joining Monaco in the summer of 2013 for a reported £3.5m.
Martial scored two goals in 15 appearances in his first season at Monaco and 11 in 42 last season.
With nine goals, he is the top scorer among the under-20 players in Europe's top five leagues since last season.
In a July 2015 study by the Soccerex 20 Football Value Index - which claimed Raheem Sterling was the most valuable young player in Europe - Martial was ranked the 17th most valuable under-21 player on the continent at £11.6m.
He's already caught the eye of France boss Didier Deschamps and clearly United think he's worth the money, but it's all about potential.
"He plays centrally, but often drifts into wide positions, more on the left than the right," said Deschamps.
