Les Monegasques are being plundered after romping to the domestic league title this past season, with many of their young players looking likely to move on.
Bernardo Silva has already been sold to Manchester City, while Tiemoue Bakayoko, Benjamin Mendy and Djibril Sidibe are also expected to leave Stade Louis II in the transfer window.
Lemar caught the eye for Leonardo Jardim's side in 2016/17, scoring 14 goals and providing 11 assists in 55 appearances across all competitions, which included a run to the Champions League semi-finals.
Gunners boss Arsene Wenger is searching for attacking reinforcements after failing to qualify for Europe's elite club competition, and is believed to have made a move for the 21-year-old.
But the Principality outfit rejected the offer and are preparing themselves for an improved bid from the north London giants, who have so far only signed defender Sead Kolasinac.