Brazil's number one for the 2018 World Cup enjoyed a fine season in Serie A and has been tipped for a move this summer, with a number of clubs interested in securing his services.
But Liverpool are expected to look at alternative targets as they are unwilling to spent £79million - Alisson's buy-out clause - to strike a deal.
The same report claims that Alisson would cost Liverpool €70 million (£61.6 million) - an extortionate amount for a goalkeeper, but a figure that could prove to be priceless if the Merseysiders go on to win the league title for the first time since 1990.
They have been linked with David de Gea and Thibaut Courtois as well as Alisson.
The Roma goalkeeper is the most sought after goalkeeper in world football after an impressive season between the sticks at the Stadio Olympico.
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'The truth is that I left everything in the hands of my agent.
Alisson wants any deal to be wrapped up before the World Cup, so he can remain as focussed as possible during the tournament.
Alisson is hoping to see his future resolved before taking in World Cup duty with Brazil, with the highly rated goalkeeper eager to avoid any distractions in Russian Federation. If everything is not resolved, it will be discussed after the World Cup.
He is likely to be Brazil's No 1 when they begin their World Cup campaign against Switzerland on Sunday. "We'll see if something happens during this week".
"There are options that we are working on, together with Roma and with respect for my club", he told Globoesporte.