The Reds have a £62m bid rejected by Roma earlier this week but Liverpool have continued talks with the Serie A giants and have submitted a second bid of around £66.8m.
That would make Alisson the most expensive goalkeeper in history and Liverpool are keen to get the deal over the line before Premier League rivals Chelsea raise their interest.
Alisson is keen on moving to Anfield and the Guardian report that Liverpool are planning for his medical at the club in the next few days.
Alisson is now expected to be left out of Roma’s pre-season tour squad in order to complete a transfer away from the Italian capital.