The 21-year-old is at the top of Arsene Wenger's summer wishlist after scoring 14 goals and recording 11 assists from the flanks last season.
Wenger was recently spotted in the French city of Nice alongside chief executive Ivan Gazidis, and the two have spent much of their summer so far working on deals for targets in Ligue 1.
That includes Monaco's star striker Kylian Mbappe, who will make a decision on his own future when he returns from holiday, and another forward in Alexandre Lacazette, with Lyon president Jean Michel Aulas expected to allow his talismanic frontman to depart if a bid in excess of £57m is received.
The Gunners have already lodged a £44m bid to Lyon for 26-year-old Lacazette which was rejected by the French side, but the signing of Bertrand Traore from Chelsea is an indication that the Ligue 1 giants are preparing for life after Lacazette, who has signalled his intentions of moving to the Premier League.
Wenger remains hopeful that a potential move for Lemar could pave the way for Mbappe to follow his compatriot to Emirates Stadium, and Goal understands that the Arsenal boss has not given up on signing the world's most sought after 18-year-old footballer this summer.
Highly-rated Lemar would compete for a place in the starting line-up alongside Alex Iwobi and Alex Oxlade-Chamberlain, although the latter has yet to sign a new deal with his representative Aidy Ward still looking to get his client the best deal possible with Liverpool and Chelsea keeping tabs on his contract situation
Lemar's arrival in north London would see him become Arsenal's second signing of the summer after the free transfer of Bosnian left-back Sead Kolasinac.
Meanwhile, Alexis Sanchez will resolve his future after returning from Confederations Cup duty with Chile. The 28-year-old has less than a year left on his current contract with Manchester City, Paris Saint-Germain and Bayern Munich all monitoring his situation, but Alexis will make a joint decision with his agent Fernando Felicevich following the tournament.