Arsenal and Liverpool both tried and failed in an effort to secure Lemar over summer, with Arsene Wenger's side making a deadline day attempt to bring him in for £92 million to replace Alexis Sanchez. Liverpool also made two bids for Lemar just days before the transfer window closed.
Monaco will likely demand at least the £92m that Arsenal offered on August 31 for Lemar, who initially rejected the move to the Emirates as Wenger's side were mulling over an offer from Manchester City for Sanchez.
Lemar's contract runs out in 2020 and it is likely that he will wait until the summer to move despite Monaco having dropped out of the Champions League and looking highly unlikely to defend their Ligue 1 title against a powerful Paris Saint-Germain side.
Goal understands Chelsea are interested in adding depth to their forward line in January after missing out on Fernando Llorente over the summer, who chose to move from Swansea City to Tottenham for £15 million. Juventus's Alex Sandro and Everton's Ross Barkley are other targets that Chelsea missed out on but retain an interest in.
The 22-year-old's fee will likely represent a transfer record for all three clubs holding an interest in him.