Liverpool's interest in Lemar broke on Monday with French publication L'Equipe reporting the Premier League side have made an offer of €72m (£67m) plus a further €8m in bonuses for the France international.
Monaco have been adamant throughout the summer they will not cash in on the 21-year-old, turning down two bids from Arsenal who have since appeared to give up on signing the attacker this summer.
But the same publication reported this morning that Liverpool were determined to strike a deal with the club and will not take no for answer despite Monaco's firm stance.
L'Equipe now understand the Reds have returned with a new offer; €50m up front with a further instalment of €30m to be paid next summer. Belgium international Origi has also been offered on a one-year loan deal.
Liverpool are now awaiting a response from Monaco vice-president Vadim Vasilyev.
Origi has managed just nine minutes of first-team football this season with the trio of Roberto Firmino, Sadio Mane and Mohamed Salah now Jurgen's Klopp's firm first-choice options. Daniel Sturridge and Dominic Solanke would appear to be the German's preferred options off the bench with the Liverpool coach recently admitting the Belgian could indeed leave the club – if new faces arrive before the transfer window slams shut.
"We will see," Klopp said after the 4-0 thrashing of Arsenal when quizzed on Origi.
"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."
While Liverpool appear determined to test Monaco's resolve, other sources in France however would suggest they might be wasting their time. Both SFR Sport and RMC Sport indicate that Monaco's position on the player hasn't changed and are adamant he is not for sale at any price.
Tottenham meanwhile tried to sign Origi in 2014 with the forward instead opting for Anfield, but appear to have revived that interest with the player's father confirming interest from the north London side.