Sky Sports claims the Reds are monitoring the 18-year-old's situation at the Ligue 1 club but are unconvinced that he wants to move to England.
Arsenal are understood to have had an ambitious £87m bid rejected by Monaco for Mbappe, while Real Madrid reportedly had a world-record £103m bid turned down for the forward.
The France international was named the Ligue 1 Young Player of the Year after scoring 15 goals in Monaco's title-winning 2016/17 campaign.
Reports earlier in the month said Liverpool boss Jurgen Klopp had held preliminary talks with Mbappe's agent.
The Reds have already signed winger Mohamed Salah for £34m ($43m) from Roma and 19-year-old striker Dominic Solanke from Chelsea.
A separate Daily Mirror claimed that Real Madrid president Florentino Perez had met with Mbappe's family to convince them to move to the Spanish capital.
Mbappe has said he would only play for a club that offered regular first-team football to boost his chances of playing for France in the 2018 World Cup.
"I will make a choice based on my sporting career, and my career is first and foremost, and I am at an age where I need to play, and that will be paramount," he told beIN Sports in May.
"I'm not 35 or 36, an age where I have to manage [my body], I still have everything to learn, this year was discovery, and now comes the most important step: confirmation.
"And I will be the sole decision-maker, and I will not go anywhere where I will be forced to go."
Monaco remain keen on holding on to Mbappe for at least one more season, with the club's chief executive saying last month that the striker would be offered a new contract.
"The plan is to do everything to prolong him and to do everything that he stays a minimum one more year with us," Vadim Vasilyev told the Daily Telegraph.