Press Association Sport understands Liverpool have agreed to meet Roma's valuation of 75 million euros (approximately £67m) for Alisson, a figure which would make the 25-year-old Brazil worldwide the most expensive goalkeeper in history.
Roma's sporting director Monchi has stressed selling Alisson "would not show a lack of ambition" as he confirmed the player was in Liverpool on Thursday.
He started all five games for Brazil at the World Cup in Russian Federation where they bowed out in a quarter-final defeat by Belgium.
Alisson has been priced up at £5.5m in Fantasy Premier League, half a million more expensive than fellow Liverpool goalkeepers Loris Karius and Simon Mignolet.
Born in Novo Hamburgo in southern Brazil in October 1992, Alisson spent his youth career with Internacional, progressing through the Brazilian giants' ranks and into their first-team squad.
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"As the negotiations were at an advanced stage, I replied to him straightaway saying, 'Calm down, I'm on my way!'".
Liverpool's new signing Fabinho has said he will try to persuade Kylian Mbappe to move to the club.
"I understand City are the champions and produced excellent football last season, but we will do our best to get the title from them and make everyone here happy". "It's the market, that's how it is and we will not think a lot about it".
The goalkeeper, who passed his club medical last night, is relatively inexperienced, but this should be cause for excitement rather than worry among Liverpool fans. I want to have them desperately but the season is really long. The Brazilian arrived at Roma two years ago but spent his first season as Wojciech Szczęsny's understudy at the Stadio Olimpico, failing to play a single minute of Serie A action before the Poland worldwide departed twelve months ago.
Asked if his former club could challenge for the Premier League title, Kewell said: "I think they are heading in the right direction".
Of the goalkeepers who made more than 20 starts in Europe's top five leagues last season, Alisson's save percentage is bettered only by De Gea and Atletico Madrid's Jan Oblak.
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