After managing to hold onto one of their prized assets during last summer's transfer window, Monaco were adamant Lemar would not be sold during the January transfer window, with vice-president Vadim Vasilyev determined to see the 22-year-old finish the season at the Stade Louis II.
Liverpool resisted the urge to bid for the France international during the mid-season window despite collecting £142m following the sale of Philippe Coutinho to Barcelona. RMC Sport reported last month the Merseyside would continue to monitor the situation heading into the summer, however.
L'Equipe also reported at the beginning of January Monaco would "soften their stance" over Lemar's future at the club, aware the player was showing little interest in extending his contract at the club, which currently runs until 2020.
The French publication, as relayed by the Mirror, now reports the attacker has now turned down a new deal from the club, despite an offer to significantly increase his rather paltry £22,000-a-week earnings at the Ligue 1 champions.
Lemar's decision to decline that offer suggests he will look to leave the club this summer, possibly after France's World Cup campaign in Russia in which he is expected to have an important role in.
Lemar, who began his career at Caen before joining Monaco in 2015, played an integral role in the club's success last season, scoring 14 times and providing 17 assists as his side won their first Ligue 1 title in 17 years while also reaching the semi-final stages of the Champions League.
Monaco have struggled to match those feats this term after a summer where they lost key figures Kylian Mbappe, Bernardo Silva, Benjamin Mendy and Tiemoue Bakayoko and are currently sat 12 points off the summit of Ligue 1 and having finished bottom of their Champions League group.
Lemar himself has also struggled to match his exquisite form from last season, with his contribution falling to three goals and five assists in all competitions this term.