The Daily Mirror is reporting that the Reds are keen to try and sign the 22-year-old France international, who they were heavily linked with a move for in the summer.
Lemar has been in decent form for Monaco so far this season, scoring two goals and making four assists in 13 Ligue 1 games for the Reds.
Liverpool FC are believed to be on the lookout for attacking reinforcements this month after they reluctantly agreed to sell Coutinho to Barcelona in a deal reported to be worth around £145m.
The same report says that Jurgen Klopp has identified Monaco star Lemar as his number one target to replace Coutinho at Anfield.
The story claims that Monaco have made it clear that they do not want to lose Lemar in the January transfer window, but Liverpool FC are poised to test their resolve given any sort of encouragement.
The Reds have already made their first signing of the January transfer window after landing defender Virgil van Dijk in a big-money deal from Southampton at the start of the month.
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