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.
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.
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?”
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