Liverpool are on the lookout for any last-minute additions to their squad before the transfer window shuts on Thursday and the Reds are targeting a move for Lemar.
The talented midfielder has been strongly linked with a switch away from the Ligue 1 champions with Arsenal showing interest in recent months.
But French newspaper L’Equipe claim Liverpool have now tabled a bid of €72m (£66.6m).
The report claims Liverpool officials will hold talks with Monaco’s vice president Vadim Vasilyev today over the possibility of a switch.
Lemar remains under contract with Monaco until 2020 but the French club could be prepared to cash in with Liverpool ready to pay a club-record transfer fee.
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Reports over the weekend suggested Liverpool had made an enquiry over his availability.
But with the deadline fast approaching the Merseyside outfit have put the wheels in motion with an official bid.
Lemar is a versatile player who can play anywhere across the frontline and usually on the left.
The France international can also be deployed as a left wing-back and would give Klopp options if he decides to experiment with his formation.
Lemar registered 12 goals and 14 assists for Monaco last season as they won the French league and reached the semi-finals of the Champions League.
Growing concerns Philippe Coutinho could be heading to Barcelona has left the Reds eager to find reinforcements with a busy campaign ahead that includes Champions League football.
Reports in Spain suggest the Brazilian is edging closer to a move to Barcelona with owners Fenway Sports Group ready to negotiate.
Lemar’s potential arrival and Coutinho’s likely departure could also have an impact on Divock Origi’s future.
The Belgian has been linked with a loan move and has played just nine minutes of Liverpool’s campaign this season.
On the possibility of signing players before the 11pm deadline and Origi leaving on loan, Klopp said: “We will see.
“That’s not something to talk about here. If players come in, I think it makes sense to think about what we do on the other side. They’re obviously involved in this (the players). I’m not sure, we will see, you have to wait.
“No games now so I hope you can enjoy the next five or six days with all the rumours and all that stuff. I’m not looking forward to it but it’s part of the game eh?”