In a shorter transfer window this summer, the deadline closes on 9 August.
Alisson is the number one choice for the Brazil National team and played in every game for Brazil at the World Cup in Russian Federation.
Alisson made 49 appearances for Roma past year, in what was his second season at the club following his transfer from Internacional.
Alisson's future at Roma has been under the microscope over the last few months and he has been widely linked with a move away from the Stadio Olimpico.
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Now, it is being claimed in the same report that AS Roma are indeed willing to let Alisson leave the Italian club this summer, handing Liverpool FC a potential boost.
The player's agent then fuelled speculation that the South American is heading to Liverpool over Chelsea after liking a photoshopped image of Alisson in a Liverpool kit on Instagram.
However, Roma are thought to be holding out for a further €5 million for the former Internacional stopper, while the Reds also face competition from Real Madrid and Chelsea for the Brazil global.