Liverpool are still understood to be considering a move for a new goalkeeper this summer following a string of errors from current number one Loris Karius, while Chelsea are looking for potential replacements for Thibaut Courtois, who is thought to be on the verge of joining Real Madrid.
Gianluca Di Marzio claims that Alisson's agent Ze Maria Neis will speak to Roma today and explain that his client wants to leave the club, should a suitable offer arrive.
Roma are yet to receive an official bid, but both Liverpool and Chelsea are now mulling over an offer after Real Madrid dropped out of the race.
Alisson's price tag is thought to be in excess of €70m (£61.9m), although Roma may be forced to lower those demands should the Brazil number one push for a move away from the club.
There have also been hints on Instagram that Alisson is willing to leave, with the goalkeeper removing Roma from his bio and his agent liking a photoshopped image of the player wearing a Liverpool top.