The Reds are desperate to land the Brazil international and are prepared to pay a world-record fee for a keeper in order to do so.
Crucially, they have received encouragement from Roma that the Serie A side are willing to sell. Liverpool had feared they would be priced out of a move for the 25-year-old, and that they may have been beaten by Real Madrid, who had also expressed an interest.
Chelsea are another club to have been credited with an interest in the goalkeeper over fears that they may lose Thibaut Courtois to Real Madrid.
There is, however, a growing confidence that a deal between Liverpool and Roma can be agreed, with the Serie A side already believed to be eyeing up replacements.
Alisson was identified as Jurgen Klopp's preferred target last season, with the Reds boss aware that none of Loris Karius, Simon Mignolet or Danny Ward have nailed down the No.1 shirt at Anfield.
Karius's errors in the Champions League final in Kiev, and doubts about the lasting effect they will have on his confidence, have reintensified Liverpool's efforts to recruit Alisson, while Ward is said to be impressing Klopp in training and could get a chance to stake his claim if a deal does not get finalised.
Now, the obstacle is agreeing a fee with Roma, who are keen to avoid a repeat of last summer when they sold Mohamed Salah to Liverpool at what now looks a ridiculously low fee.
The Italian club have made it clear that they are not prepared for the same to happen with Alisson, and Goal understands they could settle for a fee of around €75m (£67m/$88m).
The player's agent Ze Maria Neis, meanwhile, had earlier 'liked' an image on Instagram that depicted Alisson wearing a half-and-half Roma-Liverpool shirt - fuelling suggestions that a deal is not far away from completion.
There are outgoings for the Reds to concern themselves with too, with Harry Wilson all set to join Derby on Tuesday on a season-long loan deal.
Goal understands Klopp's side are also weighing up offers for Ryan Kent from both the Championship and Scottish Premiership, and Sheyi Ojo is also attracting heavy interest from English football's second tier.
Promising youngsters Rhian Brewster and Taiwo Awoniyi, meanwhile, are both set to sign new contracts this week.
Liverpool play Blackburn in a friendly on Thursday, which Ward is set to start, and fly to the USA for a three-game tour on Saturday morning.