The Reds are reportedly keen on the 20-year-old midfielder.
The 20-year-old has emerged as a prominent member of Leonardo Jardim's side following a summer switch from Caen, his 19 Ligue 1 starts yielding five goals.
It has been a trying season for the Principality side, who have had to contend with the loss of striker Anthony Martial following his £57 million switch to Manchester United back in September.
Monaco, sit third in Ligue 1, 27 points behind champions Paris Saint-Germain, were handsomely compensated over the transfer, and speculation is rife that they are set to pull in similar money for the sale of Lemar.
French newspaper L'Equipe report that Liverpool, along with Bayern Munich, Manchester City and Atletico Madrid, have acted on their interest by contacting his representatives ahead of what would be a mega-money move.
If true, manager Jurgen Klopp may be moving with the intent of replacing Joe Allen, who has, to his credit, looked excellent since the German has come in, but first-team opportunities remain scarce for the Welshman on the whole.