The 22-year-old, who has proven himself to be a highly talented individual, is under contract at Old Trafford until 2019.
RMC says Martial is understood to have turned down a new five-year contract offer, after becoming exasperated with his lack of starts under Jose Mourinho.
Summer signing of Romelu Lukaku and January acquisition of Alexis Sanchez have hampered his game-time further.
Juventus have been made aware of the situation, and are keen to do a deal, having been informed that Martial would likely cost around £60m.
Generally perceived as a player that has not bought into Mourinho's approach to the degree that the Portuguese knowingly demands of his players, it is undoubtedly with some reluctance that United fans would see Martial leave.
Juventus, Tottenham, Chelsea and Paris Saint-Germain are all monitoring Martial's situation, with United ready to cash in on him if he can not be persuaded to stay.
With Mourinho generally favouring consistency amongst his attacking players, the tendency with which Martial blows hot and cold may well see him sacrificed in favour of a more suitable candidate.