Arsenal made a £90m (€100m) bid to try and sign 22-year-old Lemar on the final day of the summer transfer window as they looked to bring in a replacement for Alexis Sanchez, who was then on the brink of joining Manchester City.
That followed two bids from Liverpool during a frenetic final week, one of which was said to have been worth €90m lodged 48 hours before the window closed.
Interest from Arsenal and Liverpool re-emerged in late 2017 with L'Equipe report both clubs would make new bids in the January window – adding Premier League champions Chelsea had also been in contact with the France international's representatives to make their interest known.
Liverpool's renewed efforts to sign the attacking midfielder come as Philippe Coutinho edges closer to an Anfield exit, with the Mirror reporting the Merseyside club have identified Lemar as a direct replacement for the Brazilian playmaker.
Lemar remains on Chelsea's radar despite the seemingly imminent signing of Ross Barkley from Everton, but both the Blues and Liverpool may have to wait until the summer before any deal can be struck.
"We are not in the habit of doing major sales of players (in the January window)," Jardim said, adding his Monaco players "understand" the club's position.
Lemar was one of the new star players retained by Monaco after the club was raided during last summer's transfer window. Benjamin Mendy, Bernardo Silva and Tiemoue Bakayoko, all integral to the club's Ligue 1 title success joined Manchester City and Chelsea respectively while the jewel in the club's crown Kylian Mbappe joined Paris Saint-Germain on loan in a deal which will be made pemrnanent at the end of the season for a £166m fee.