While Monaco are adamant the France global is not for sale this summer, the Merseyside club appear determined to test their resolve and have offered €80m plus Divock Origi on a one-year loan deal to try and tempt them into doing business before Thursday's transfer deadline.
Arsenal were chasing the attacker all summer but have given up landing the star as he price continues to rise but the Mailreport the Kop side have made the massive bid.
Lemar has attracted attention from Arsenal, Manchester United and Liverpool.
It is also understood the Ligue 1 champions want Lemar to stay with them beyond this summer's transfer window.
And Liverpool have also offered to include Origi in negotiations, with the young striker potentially spending the campaign on loan with Monaco.
According to the Liverpool Echo, corroborating claims in French publication L'Equipe, they will now offer £75 million for the youngster.
Arsenal are prepared to meet a £75m buyout clause to sign Real Madrid's Spain midfielder Marco Asensio, 21, before Thursday's transfer deadline.
Is Coutinho Barça deal Lemar linked?
"If players come in, I think it makes sense to think about what we do on the other side", he said following his side's 4-0 embarrassment of the Gunners on Sunday afternoon.
"But we have to do the right things, look at the right players and negotiate with the right clubs".
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.
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