The France international has become one of the most in-demand players on the market, seeing his name linked with an array of clubs across the continent.
Arsenal and Liverpool emerged as the rumoured frontrunners to sign Lemar both last summer and in January, but Monaco ultimately blocked a move from going through.
According to Le 10 Sport, however, the Reds have made a fresh approach for the 22-year-old and are now on the brink of reaching an agreement with the Ligue 1 champions.
The report indicates that Liverpool have "made some good moves" in recent weeks and are now in the final stages of negotiations.
Lemar has netted two goals in the French top flight this term and chipped in with a further seven assists.