Reds boss Jurgen Klopp has already spent the best part of £100million since the end of the season to bring in Lemar's Monaco team-mate Fabinho and fellow midfielder Naby Keita from RB Leipzig.
However, the German tactician is keen to further bolster his squad in the transfer window and has also been heavily linked with a move to bring in Nabil Fekir from Lyon.
But fresh reports are claiming that Klopp will move for his fellow France international, Lemar, who is valued at £100.5m by the Ligue 1 outfit.
The 22-year-old, who has scored 22 goals and provided 32 assists in 127 games in all competitions for Monaco, is attracting the interest of several top sides including the Merseysiders.
But the Principality outfit are keen to hold on to him having offloaded several star players in the past 12 months.