Both Arsenal and Liverpool made bids for Lemar during the final week of the summer transfer window with the Gunners coming close to clinching a €100m (£90m) deal, only for the move to fall through at the eleventh hour.
Reports in France in December suggested both clubs would again test Monaco's resolve with offers for the attacking midfielder during the January transfer window, adding Chelsea had also been in contact with the players' representatives to make their interest known.
Monaco, who trail Ligue 1 leaders Paris Saint-Germain by nine points, have been resistant to the idea of cashing in another of their prized assets during the mid-season window with manager Leonardo Jardim recently insisting his players "understand" the club's position.
But a report from L'Equipe on Saturday claimed that Monaco have softened that stance, believing they can now earn more from his departure now than if they wait for the summer.
Confirmation of Philippe Coutinho's move to Barcelona has intensified suggestions Liverpool will look to Lemar to fill the void left by the Brazil international. But according to Sky Sports, the Merseyside club are not looking to bring forward a possible move for the 22-year-old.
That development will be welcomed by Arsenal if Arsene Wenger does decide to push forward with a move for Lemar. Manchester City still hope to strike a deal for Alexis Sanchez this month after their summer deadline day move collapsed, a development that would once again see Arsenal turn to Lemar as a replacement for the Chile international, Saturday's report from L'Equipe added.
In September, Wenger confirmed the club would return for the player at a later date and while the Gunners boss dismissed suggestions an offer has already been made at a recent press conference, reports that Monaco might now be ready to sell could prompt another bid.