Liverpool have had two bids rejected by Monaco for winger Thomas Lemar, according to Sky sources.
Sky Sports News understands the latest offer made by Liverpool was around £64.8m, having previously submitted an offer worth £55.5m.
It is also understood Monaco want Lemar to stay at the club beyond this summer's transfer window.
Liverpool manager Jurgen Klopp last week expressed his desire to strengthen his squad further after his side secured qualification to the Champions League group stage.
Mohamed Salah, Andy Robertson and Dominic Solanke have been the only new arrivals at Anfield so far this summer, while long-term target Naby Keita will join next summer for a club record £48m fee.
Speaking after their 6-3 aggregate win over Hoffenheim last week, Jurgen Klopp said: "It's [Champions League qualification] a big influence on the transfer market, especially if we do it more often.
"I said I think before the game, if you talk to a player, they really often [mention it]... especially players we talk to."
Lemar, who helped Monaco win the French title last season for the first time in 17 years, was in France's starting line-up for the friendly victory over England in June.
The promising 21-year-old scored 12 goals in 39 appearances last season and has three years left on his current contract.
Arsenal had been in talks to sign Lemar all summer and are believed to have had a £30m offer rejected last month while Barcelona were also understood to be tracking the player.
Monaco have already sold a number of key players this summer, including Bernardo Silva and Benjamin Mendy to Manchester City and Tiemoue Bakayoko to Chelsea.
They said last month there would be 'no more departures' amid continued interest in Lemar, Djibril Sidibe and midfielder Fabinho and Kylian Mbappe. The latter is now understood to be on the verge of a season-long loan move to Paris St Germain.