Liverpool have made a £55.5m bid to Monaco for Thomas Lemar, Sky Sports News understands.
It is believed Monaco have yet to accept or reject the offer as they assess options in their squad.
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Arsenal had been in talks to sign the attacking midfielder 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 maintained Lemar is not for sale, with the Ligue 1 champions having already sold a number of key players 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 Klian Mbappe. The latter is now understood to be on the verge of a season-long loan move to Paris St Germain.
Mohamed Salah, Andy Robertson and Dominic Solanke are the only new arrivals at Anfield this summer, but Jurgen Klopp hinted that could change after they secured their passage to the Champions League group stages.
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 Monaco contract.