The Reds have been in intensive talks with the Serie A club this week, and now have permission to speak to the player about a move to Anfield.
The deal would make the 25-year-old the most expensive goalkeeper in history.
Alisson, it is understood, is open to the idea of a move to Merseyside, but Liverpool could face competition from Chelsea, who look set to lose their own goalkeeper, Thibaut Courtois, to Real Madrid.
Liverpool, though, are growing confident they can win the race, which would mean they have broken the world transfer record for both goalkeepers and defenders in the last seven months.