The France international was wanted by the north London club in the summer transfer window, and the Premier League outfit came close to completing a club-record €100m (£87.8m, $116.6m) transfer on deadline day. However, the deal did not go through.
Arsene Wenger confirmed Arsenal's €100m bid for Lemar and revealed that the player wanted to continue at Monaco, which convinced the winger to snub the Gunners' approach.
However, the French manager confirmed that his side will return for his compatriot in the future.
According to The Telegraph, Barcelona have also identified Lemar as a transfer target for the next summer transfer window, with the Catalan club set to rival the Emirates outfit in their efforts to sign the Ligue 1 attacker.
Liverpool were also interested in signing the Monaco forward in the summer. The Reds made an approach to bring Lemar to Anfield, and could join Arsenal and Barcelona in the race to sign the 21-year-old next summer.
The Mirror reports that Arsenal are in pole position to land Lemar and have identified him as the player to replace Alexis Sanchez at the Emirates. The Chilean has less than a year left on his contract and is expected to leave the club on a free transfer after the end of the season.
With interest from Arsenal, Liverpool and Barcelona, Lemar is set to make a decision on his next possible destination soon. He is also aware that he can command a weekly wage of £250,000-a-week before deciding his future club.
Monaco vice-president Vadim Vasilyev has already confirmed summer approaches from Arsenal and Liverpool, while also admitting that the club are preparing for Lemar's exit after the end of this campaign.
"Liverpool were really interested. But it was Arsenal who were really close to finalising the deal," Vasilyev told Sky Sports.
"It was quite close. At the same time, it was just too late, not enough time to really follow up on this deal. Not all the elements of the puzzle fell in place. I'm very happy he stayed with us. For us, we need a player like Thomas Lemar. He's very important to our team.
"Definitely, next summer will be something to look at, although you know in football there are no guarantees. We had some players who were supposed to leave this summer but didn't leave because of X, Y, Z. I think it would be fair to say Thomas will have a fair chance to an exit next year."