The Merseyside outfit were in pursuit of the France international in the closing stages of the summer window, but a deal never transpired.
Arsenal also made a last-ditch attempt to sign the £90m-rated midfielder, but the Ligue 1 giants refused to sell after allowing a number of key players to leave in that window.
According to the Liverpool Echo, Reds manager Jurgen Klopp has not given up on Lemar, and club officials have already made contact with Monaco in a bid to open talks.
The report claims that Arsenal remain interested, while Chelsea and Manchester United are also keeping tabs on the player, but it is believed that he finds the prospect of working under Klopp appealing.
A space in the Liverpool team may open up for Lemar, amid widespread reports that Philippe Coutinho could be on his way to Barcelona.