Ligue 1 champions Monaco have already turned down a £60m offer for the 21-year-old.
Lemar scored 14 times last season as his team reached the Champions League semi-finals and won their domestic title.
But Liverpool insist their move for the France international is not linked to the future of playmaker Philippe Coutinho.
The Reds remain adamant they will not sell Coutinho, having rejected three bids from Barcelona this summer, including an offer of £138m.
Countinho hasn’t played for Liverpool this season due to injury.
But he trained with Brazil yesterday seemingly without any hint of the back injury and illness which have ruled him out of Liverpool’s opening five games of the season.
Friend and surgeon Dr Michael Simoni claimed on Monday that Coutinho’s injury was the result of stress over his potential move to Spain.
“He’s a bit anxious about the possibility of changing club,” he said ahead of Brazil’s World Cup qualifier against Ecuador on Friday.
“Of course we talked about this possible transfer, about the things that are in his heart right now. His back pains, in my opinion, are explained by this anxiety. Muscular tension normally does that.”
While Liverpool are refusing to sell their star man, Southampton are just as adamant that Virgil van Dijk will not be heading to Anfield ahead of tomorrow night’s transfer deadline.
Van Dijk hasn’t played this season and has been training with the under-18s after having a transfer request refused.
The £70m-rated defender wants to join Liverpool while Chelsea and Arsenal are also interested.
But Saints are standing by vice-chairman Les Reed’s public pledge that Van Dijk will not be sold.
Meanwhile, Naby Keita says he will be a long-distance Liverpool fan for the next year.
The Guinea international midfielder has agreed a £60m move from RB Leipzig but won’t join his new club until next season.
“Until I join my new club, next summer, I will remain an interested supporter from a distance,” said the 22-year-old.
“I am delighted that an agreement has been reached which will allow me to join Liverpool Football Club next summer, when I will become part of a project that excites me greatly.
“Having my future resolved means I can now focus on helping RBL achieve great things this season.
“My commitment to RBL remains absolute for the remainder of my time at the club. I have given my all every time I have worn the shirt - that will be the case until the final whistle, in my final appearance.”
