The Reds are attempting to pull off a deal before the transfer deadline on Thursday night.
Monaco have resisted selling Lemar all summer amid interest from Arsenal but Liverpool are set to test their resolve by making the biggest offer yet.
The Reds are prepared to offer Divock Origi in as part of the deal, which could reduce the cash outlay.
Monaco have signed Keita Balde and Stevan Jovetic during the past 24 hours, who could be potential replacements for Lemar and Kylian Mbappe.
Lemar scored 14 goals for Monaco last season and also provided 14 assists, earning a call up into the France international squad, featuring against England earlier this year.
Liverpool have already broken their transfer record twice this summer to sign Mohamed Salah and Naby Keita, who arrives in 2018.
A move for Lemar will require them to break it again but the Reds are showing they have the ambition to do so, even if they have left it daringly late.