The 21-year-old has been heavily linked with a move to the capital all summer and the player looked set to join compatriot Alexandre Lacazette at the Emirates.
However, although the club are still attempting to push through a deal, Monaco do not want to lose their winger due to the number of departures at the club.
Tiemoue Bakayoko joined Chelsea at the weekend, while Benjamin Mendy is expected to join Bernardo Silva at Manchester City.
The Ligue 1 champions are also doing all they can to keep hold of Kylian Mbappe, who has been linked with a move to the Gunners, Real Madrid and Paris Saint-Germain.
Although Lemar wants to join the Gunners, Arsenal are growing less optimistic that the club can complete a deal for the French international.
That news has come from BBC Sport journalist David Ornstein, who has become a cult hero amongst some of the fans for his record of announcing transfer news regarding the club.
As well as revealing the news about a Lemar deal looking unlikely, Ornstein also commented that the club’s top priority in the transfer window is to keep hold of Alexis Sanchez.
The Chilean, who has one year left on his current deal, is reportedly keen to leave the club in the summer after three seasons in north London.
Sanchez’s Arsenal future looks in doubt, especially given his interview with Radio Sport Chile at the weekend stating that he wants to win the Champions League.
While the Gunners will be playing in the Europa League next season, Wenger will remain hopeful of convincing the former Barcelona man of extending his contract at the club.
Who would you prefer to see join Arsenal – Thomas Lemar or Riyad Mahrez? Let us know in the comments below.