Monaco have so far rejected two offers from the Gunners but Arsene Wenger is determined to bring the 21-year-old to the Emirates.
Sky Sports claim talks are ongoing between the two clubs.
Arsenal broke their transfer record earlier this summer when they spent £46.5m on Alexandre Lacazette.
Wenger wants to add quality to his side as they seek to challenge for the Premier League title.
Lemar was impressive as Monaco won the French title and made it the semi-finals of the Champions League last season.
The forward netted 14 goals and provided 17 assists to his team-mates in all competitions.
And Arsenal hero Robert Pires thinks Lemar would be a great addition for the Gunners.
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He said: “I love him as a player. For me, Thomas Lemar is one of the best French players there is.
“He can play on the left, he can play on the right, he can score goals and he can make assists. Lemar is the kind of player who can make a difference.
“He has the right profile to play for Arsenal, definitely. I hope for him and for Arsenal that it happens.”
Arsenal start their Premier League campaign against Leicester on Friday, August 11.
