Jose Mourinho has been keen to add another attacking option to his squad this summer, explaining in July he wants "another attacking player, especially [one] that can play through the wings to give me more attacking options."
Inter Milan winger Ivan Perisic was thought to be club's first choice for the role but the Italian side's refusal to listen to any offers that fall below their £48m valuation of the player has so far prevented a deal being struck.
The Croatia international is now determined to remain at Inter, according to his manager Luciano Spalletti, with a new contract likely to be offered in the coming weeks.
United have also given up in their attempt to prise Gareth Bale away from Real Madrid with Mourinho acknowledging the former Tottenham Hotspur star has no desire to leave the Spanish capital.
But according to the Daily Record, Mourinho has added 21-year-old Lemar to his list of attacking targets. In June, L'Equipe listed United as one of seven clubs interested in signing the France international but suggested Arsenal boss Arsene Wenger determined to sign him "at all costs."
The Gunners have seen bids of £35m and £40m rejected by the club however with Monaco's hierarchy determined to ensure there are no more high profile departures from the club following Bernardo Silva's move to Manchester City and Tiemoué Bakayoko's move to Chelsea.
But United could now try test their resolve with just two weeks of the transfer window remaining.
Lemar played an integral role in Monaco winning their first Ligue 1 title in 17 years and reaching the Champions League semi-finals last term, collecting a superb haul of 14 goals and 17 assists in all competitions.