The 22-year-old France international could be sold in the current transfer window, with Arsenal and the Reds both keen on acquiring his services
Monaco head coach Leonardo Jardim accepts he may be unable to keep Thomas Lemar at the club during the January transfer window.
Vice-president Vadim Vasilyev and Jardim have previously been clear that Lemar and other Monaco stars including Fabinho will not be allowed to leave mid-season.
Kylian Mbappe, Bernardo Silva and Benjamin Mendy departed last season's Ligue 1 champions, who sit second in the table, nine points behind Mbappe's new side Paris Saint-Germain.