The French midfielder is a top target for Arsene Wenger.
But Monaco don’t want to sell their star, and Arsenal have failed to tempt them into a sale.
BBC reporter David Ornstein expects Lemar to stay with his club.
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“It’s very unlikely Arsenal will sign him now,” he said on Radio Five Live.
“There is interest there but Monaco have made their stance pretty clear that he’s not for sale and it would take an incredible amount of money for Arsenal to prise him away.”
The Gunners have signed Alexandre Lacazette and Sead Kolasinac so far this summer.
And Ornstein says they could yet sign one more player.
“There may be one that Arsenal are possibly looking to bring in,” he added.
“I don’t think there is a specific name, but there is perhaps scope to bring in one more player.
“They’ve already spent quite heavily. Alexandre Lacazette - the record signing - and Sead Kolasinac was a free, but with their salaries that brings that spend up.
“They may look to bring a player in before the deadline but the priority is to shift a player out.”
