Arsenal have already made significant additions to their first-team this summer by landing Sead Kolasinac on a free transfer from Schalke and paying a club-record fee to bring Alexandre Lacazette to the club from Lyon.
Lemar has been talked up as a target for the Gunners, with Wenger a fan.
It is now being reported this morning that Wenger has returned to Monaco with a third bid of £45million for Lemar, and with the player himself desperate to come to the Emirates the French worldwide is hopeful that the deal could be finalised before the weekend.
The newspaper says Arsenal are now close to agreeing a £45m deal following a breakthrough in the negotiations and Lemar is already believed to have agreed personal terms over his move to the Emirates.
Lemar would be the ideal replacement for Alexis Sanchez as he can play on the left wing, Arsenal might also use the Frenchman in central midfield as well.
Arsenal are not giving up hope that they can persuade Sanchez to sign a new contract.
And, the clearout at Monaco has been significant this summer, with the likes of Benjamin Mendy, Bernardo Silva and Tiemoue Bakayoko all leaving.
Lemar featured for Monaco in their last pre-season game at Benfica and will hope to get the green light to complete his switch to the Premier League before they face Paris-Saint Germain in Morocco this Saturday in the Trophee des champions.
Lemar scored nine goals and registered 10 assists in 34 league appearances last season and found the back of the net twice in the Champions League.