Italian newspaper Corriere dello Sport, as quoted by Metro, is reporting that the Serie A club are determined to try and keep hold of their shot-stopper this summer and are not willing to let him leave for less than €60m (£52m).
That price tag would make Alisson the most expensive goalkeeper of all time were he to leave AS Roma for that fee this summer.
According to the same report, both Liverpool FC and Arsenal are interested in signing the goalkeeper this summer, while Real Madrid have also been credited with an interest in the Brazilian.
Liverpool FC have been linked with a move for Alisson following the less-than-convincing displays of Loris Karius and Simon Mignolet between the posts this term.
Arsenal are also thought to be in the market for a new goalkeeper as veteran shot-stopper Petr Cech approaches the end of his career.
Meanwhile, Liverpool FC will return to Premier League action on Saturday lunchtime with a trip to Crystal Palace in the top flight.
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news