Over the past five months, a number of Premier League sides have been linked with the French international, who is said to be considering his options ahead of next season.
Reports have suggested that Liverpool and Arsenal have been the frontrunners for Lemar's signature but according to Yahoo! Sports, the Merseyside giants have strengthened their position in the race for the 22-year-old.
The report claims that Liverpool have reached an agreement with the player regarding a switch to Anfield, although the English side allegedly remain short of Monaco's asking price of €100m (£87.51m).
Lemar has not replicated his form of last season, but the attacker still has two goals and five assists from 16 games in Ligue 1.