Manchester United are claimed to be keen to sign Lemar.
The France international has been the subject of speculation all summer long, with Arsenal pushing hard for his signature.
Manchester United though are claimed to be considering making a bid for the attacker.
United have become frustrated in their efforts to sign Inter Milan's Ivan Perisic, while Gareth Bale is also not for sale from Real Madrid.
Monaco have been keen to keep hold of Lemar, but United could be willing to offer a fee beyond anything Arsenal have been able to try and tempt them with.
Lemar can play on the wing and through the middle, and was a key player in Monaco's run to the Champions League semi-finals last season.
He provided 14 goals and 14 assists, having stepped up since Anthony Martial was sold to Manchester United in 2015.