The 25-year-old has been on the Premier League club's radar for a number of weeks and, on Wednesday, multiple reports claimed that a move to Anfield had been agreed.
Alisson is believed to have flown to England on Wednesday night and undergone a medical the following day, after which he returned to Italy to resume his holiday.
On Thursday night, Roma tweeted a highlight reel of Alisson's saves in Giallorossi colours alongside a message that read: "Thank you Alisson Becker and good luck with your new adventure."
Liverpool, who are yet to confirm the deal, have reportedly paid £67m for the Brazilian stopper's services on a five-year contract.
Real Madrid and Chelsea are also believed to have held an interest in Alisson, who played for the Selecao at the 2018 World Cup in Russia.