According to David Maddock of the Daily Mirror, the Reds are not prepared to part with the asking price Roma are demanding for the Brazil worldwide, which is said to be around the £70 million mark.
Before the Champions League final - and Loris Karius' blunders - reports anticipated that the Liverpool boss would keep his compatriot as the club's number one stopper, and now those claims are being made again.
It is also understood that third-choice goalkeeper Danny Ward could be promoted to compete with Karius.
Signing for £37 million and then going on to break the record for most goals scored in a 20-club Premier League season must surely classify him as one of the best deals in English football.
Since being given the starting berth at the Stadio Olimpico on a regular basis he's blossomed.
Alisson has carried on that form after the campaign during worldwide duty with Brazil.
According to Maddock, Roma are wary of losing another star player to Liverpool for a bargain price.
Liverpool bought Mohamed Salah last summer for a cut-price as AS Roma had to meet FFP regulations or face transfer sanctions and Gabriele Marcotti believes this will affect any potential negotiations for Alisson.
Some wouldn't want the club to be quite so careful.