The 25-year-old left John Lennon Airport on a private plane headed for Rome on Thursday evening. Sky in Italy say Alisson has already agreed a five-year deal.
He will become the world’s most expensive goalkeeper after Liverpool’s offer surpassed the £34.7m Manchester City paid for Ederson in 2017.
The deal was too good to turn down, according to Roma sporting director Monchi.
“We haven’t finalised anything yet [with Alisson],” he said on Thursday morning. “But it is true that the negotiations are advanced.
“When a very substantial offer comes in, you have to consider it. We weighed up the pros and cons and decided to speak to Liverpool.”
He added: “Selling Alisson would not show a lack of ambition. For me, ambition is about doing the right thing after thinking everything through.”
Alisson made 49 appearances for Roma last season, and played in both legs of the 7-6 aggregate defeat to Liverpool in the semi-finals of the Champions League.
Liverpool have now agreed to sell goalkeeper Danny Ward to Leicester City for £10m, according to Sky sources.