Arsene Wenger has been chasing the Monaco winger all summer but the Gunners have struggled to land a deal.
Emirates chiefs may launch another attempt for his signature in the coming days.
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And French outlet Le 10 Sport report that the Ligue 1 champions could sanction his departure in the final days of the transfer window.
Lemar established himself as one of Europe's hottest talents last term, providing 14 goals and 17 assists in 55 appearances.
Wenger previously admitted that Lemar has popped up on the club's radar.
The Arsenal boss said: “He's a player that we have looked at, but the rest is only speculation."
Monaco have sold a string of last season's heroes including Tiemoue Bakayoko, Bernardo Silva and Benjamin Mendy.
Vice-president Vadim Vasilyev insisted last month that Lemar would not be joining the departures list.
He said: "We have kept all of the key players that we intended to keep.
"And I will address speculation relating to Lemar - he is staying with us. He is an immense talent and is essential for the team.”
Le 10 Sport claim that Monaco could sell Lemar, Kylian Mbappe and Fabinho before the end of the month.
Manchester United are also interested in Lemar.
Jose Mourinho is considering a swoop after failing to land Gareth Bale from Real Madrid and Inter Milan’s Ivan Perisic.
Both clubs have until 11pm on August 31 to strike a deal for Lemar before the transfer deadline.
United are 6/1 with bookies BetStsars to sign Lemar.
