That’s according to French newspaper L’Equipe, who claim the Reds raised their offer for the Monaco star from £74m to £82m within a week.
Lemar, 21, shot to fame last season as Leonardo Jardim’s men won the Ligue 1 title and reached the semi-finals of the Champions League.
And this summer he was heavily linked with a move to the Premier League, with Arsenal long seeming like his most likely destination.
Liverpool emerged as an interested party earlier this week, with Lemar seen as a potential replacement for No.1 Barcelona transfer target Philippe Coutinho.
And L’Equipe claim they were willing to pay £82m, not including add-ons, to get him on-board.
L'Equipe claim that bid came off the back of an earlier proposal of £74m, plus £8m in add-ons, being knocked back.
However, Monaco chief Vadim Vasilyev opted to reject Liverpool’s second offer having seriously considered it.
Vasilyev’s decision to turn down Liverpool’s late bid may have come as a hefty blow to Lemar.
L’Equipe say Monaco’s key men had been informed earlier this year that they’d be allowed to leave if an “exceptional offer” came along.
They also suggest that the France international was only interested in joining Liverpool if he was to move.
That stance, which would have involved snubbing Arsenal, was apparently motivated by primarily sporting reasons.
L’Equipe hint the opportunity for Lemar to link up with Liverpool could present itself in January though.
