Real are in the market to replace Keylor Navas, while Liverpool have long been linked with the Brazilian as a more reliable choice than incumbent Loris Karius. The need for someone new was only accelerated by the German's costly errors in the Champions League final.
According to Globoesporte in Brazil, Roma initially wanted €100m (£87.6m) for Alisson, whose representatives have reportedly been in contact with Liverpool since December.
That would be a huge new world record for a goalkeeper, smashing for the €52m fee that took Gianluigi Buffon to Juventus back in 2001, and well over double what Manchester City paid for Ederson, Alisson's national team understudy, last summer.
However, it is explained that in reality there is hope that Roma would accept a figure of around €50m, a number still in world record territory but not totally prohibitive.
What Roma might argue, however, is that Liverpool have just agreed a deal worth up to €50m for Fabinho, a player not deemed worthy of a place in the national team. Therefore, if Fabinho is worth €50m, should Alisson, a key national team player, not be worth much more?
Budgetary constraints would be more of an issue for Liverpool than Real, who also hold the allure of being the biggest club in the world after a third successive Champions League title.
The lack of a coach at Real following Zinedine Zidane's recent departure is billed as a problem for Real, though. How willing would a player be to join them without knowing beforehand the identity of the coach they are going to be working with?