The Mirror says the Reds could have captured the 25-year-old, who joined last week for a record £63m, in July 2015.
And they would have landed the Brazil No.1 for a bargain price, after being offered a deal which would have seen them pay just £3.1m up front with £900,000 in add-ons.
UK-based intermediaries working on behalf of the player's agent, Ze Maria Neis, were asked to secure a move to England that summer when he was with Brazilian club Internacional.
Liverpool were one of several Premier League and Championship clubs who were sent a detailed scouting dossier outlining Alisson's potential.
The prospectus also included a written testimonial from Brazil's World Cup-winning keeper Claudio Taffarel hailing Alisson as the best young keeper to come out of South America.
With his reputation on the rise, Internacional were ready to sanction a sale if their asking price was met.
Alisson told his advisers that England was his preferred destination. But there were no takers – despite assurances that the player's German ancestry made him eligible for a European Union passport.