The 19-year-old, who has been linked with Arsenal, was allowed to leave the France squad on Monday to travel for a medical with Louis van Gaal's side.
Martial scored eight goals in 31 Ligue 1 appearances last season.
If Martial sing, he will become Manchester United's third most expensive signing, after £59.7m Angel Di Maria and £37.1m Juan Mata.
Spain midfielder Mata joined from Premier League rivals Chelsea in January 2014, while Argentina winger Di Maria moved from La Liga giants Real Madrid for a British record fee in August last year.
After Manchester United lost 2-1 to Swansea on Sunday, Van Gaal said he would not panic buy before the transfer window closes in England at 18:00 BST on Tuesday.
France Under-21 international Martial was called up to the senior national team for the first time on 28 August.
What you should know about Anthony Martial
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
With nine goals, he is the top scorer among the under-20 players in the European top five leagues since last season.
"Anthony Martial asked France coach Didier Deschamps for permission to travel to England to sign a contract with Manchester United," a statement from the French Football Federation confirmed.
Martial started his career at Lyon, making his professional debut in December 2012, before joining Monaco in June 2013.
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| Martial scored two goals in 15 appearances in his first season at Monaco and 11 in 42 last season |
This summer he agreed a one-year extension to his previous deal, taking his contract up to 2019.
At the time, Monaco vice-president Vadim Vasilyev said Martial was "untransferable", with Tottenham reported to have shown interest.
'Pace and power'
On calling up Martial to his France squad, Deschamps said: "He plays centrally, but often drifts into wide positions, more on the left than the right.
"He's got a good combination of pace and power. He's a young player obviously, but he's got an interesting profile in an area where it's not easy to find players with both strength and speed.
"He's got good potential and I want to see how he does with the squad."
Source: Bloom Gist Sports Desk
At the time, Monaco vice-president Vadim Vasilyev said Martial was "untransferable", with Tottenham reported to have shown interest.
'Pace and power'
On calling up Martial to his France squad, Deschamps said: "He plays centrally, but often drifts into wide positions, more on the left than the right.
"He's got a good combination of pace and power. He's a young player obviously, but he's got an interesting profile in an area where it's not easy to find players with both strength and speed.
"He's got good potential and I want to see how he does with the squad."
Source: Bloom Gist Sports Desk
