Martial, 22, is believed to be frustrated with his role at Old Trafford, having started less than half of Uniteds Premier League fixtures this season.
The arrival of Alexis Sanchez from Arsenal has not helped Martials cause and his representatives have been exploring the possibility of a summer move.
Top-flight rivals Arsenal and Tottenham and La Liga giants Barcelona are said to be interested in signing Martial, but they will face competition from Chelsea.
The Daily Telegraph say Chelsea are one of a number of clubs to have expressed interest in securing a deal for Martial, with Roman Abramovich keen to spend big after a difficult season.
Rather than bolstering the squads defensive options, Abramovich is looking to sign forwards with Martial firmly on his radar.
Martials current contract at United will expire next summer and Mourinho hopes to be able to convince him to commit to the club by signing a new deal.
But recent reports say the Red Devils boss is prepared to sell Martial if he requests a move at the end of the season.
To raise funds for the clubs summer transfer plans, Abramovich is expected to sell Michy Batshuayi, Kurt Zouma and Kenedy.
England midfielder Danny Drinkwater could also leave the club after just one season – West Ham are keen to offer him the chance to revive his career in east London.
Mourinho is believed to have identified Bordeaux star Malcom, Monaco ace Thomas Lemar and Ajax teenager Justin Kluivert as potential replacements for Martial.