Sky Italia has reported the Reds have begun preliminary talks with the Serie A club over the Brazilian.
An approach for the 25-year-old was made in January, and Liverpool are hopeful they can complete a deal within the next few weeks.
The Reds flexed their financial muscle by forking out £70m for Virgil van Dijk last month. And they may have to do the same if they want to sign their top goalkeeper target.
The Giallorossi rate Alisson as one of the world's best in his position and are looking for a world-record £62million transfer fee.
Alisson has 22 caps for Brazil and is set to feature for the Selecao at this year's World Cup.