Sky Sports say the latest offer was around £65m.
Arsenal have been repeatedly linked with the influential winger.
But Monaco, who have already seen several high-profile stars depart this summer, are keen to retain his services.
Gunners chief Arsene Wenger admitted defeat in the race for his signature last week.
He said: “It’s dead because Monaco have closed the door definitely.
“They are still on cases like Mbappe and Fabinho.
“They’ve already lost Mendy, Bernardo Silva and Bakayoko, so Monaco is at a stage where they have closed the door.”
Get French Football News reckon Monaco would be willing to cut a deal with Liverpool.
However, it’s suggested they want at least £93m to part with their prized asset.
Lemar is under contract with Monaco until 2020.
He was on fire last term, providing 14 goals and 17 assists in 55 appearances.
Lemar arrived at Clairefontaine earlier today to link up with the France squad.
