The deal is not finalised yet, but Roma is reportedly considering the world-record bid, reported Sports24.
The 25-year-old, who started his International career with sports club Internacional in Brazil, is with Roma for past two years and has played 37 Serie A games last season.
The bid report came after current Liverpool goalkeeper Loris Karius failed to hold on to a free-kick in a friendly match against Tranmere Rovers at Prenton Park. Liverpool, however, managed to defeat Tranmere Rovers 3-2 in a close contest.
Karius is under scanner after he gave away two goals in a match against Real Madrid in this year's Champions League Final which Liverpool lost 1-3. It was found later that the miss was a result of a concussion.
Alisson played for Brazil in the recently concluded 21st edition of the FIFA World Cup where he appeared in all the five matches of the team. The team was knocked out of the tournament after they suffered a 1-2 defeat at the hands of Belgium in the quarter-final clash.