The Gunners have had two bids rejected by the Ligue 1 champions, who are holding out for a fee in the region of £50m.
Lemar has been a target for Arsene Wenger all summer and is his key priority before the transfer window slams shut.
According to Sky Sports, talks between the two clubs are on-going in a bid to agree a deal.
Lemar enjoyed an incredible season for Monaco last term, providing 14 goals and 17 assists in 55 appearances.
Monaco have publicly stated that Lemar is going nowhere after selling a number of their star men including Bernardo Silva, Benjamin Mendy and Tiemoue Bakayoko this summer.
“All the players who were important last season are important to keep for the club,” manager Leonardo Jardim explained.
“But it has been shown over the years that it's not always possible to keep the players, and you know how our project works.
“So our goal is to work well with players who will stay here after the transfer window closes.”
It was reported earlier this week that Liverpool could consider a move for Lemar if they lose Philippe Coutinho to Barcelona.
