Gianluca Di Marzio has reported that Alisson is "close" to a move to Anfield, while Italian football journalist Fabrizio Romano also reported the Reds have started talks with the player:
According to Gazzetta dello Sport (h/t Sport Witness), the Reds have put forward the "most appropriate offer" at this juncture, although their package worth a potential €70 million is still said to be below Roma's €80 million valuation of the Brazil international.
Gazzetta first reported the prospects of each club signing the player, with the Reds said to have a 45 per cent chance. Chelsea's hopes are marked at 35 per cent, with Real Madrid at 10 per cent and a Roma stay also 10 per cent.
It's added that an agreement between Roma and Liverpool "isn't far" away, and Real Madrid appear unlikely to be serious contenders for the goalkeeper given they are chasing Chelsea stopper Thibaut Courtois. That's despite the fact Alisson is said to prefer a move to Madrid.
With international duty now over for the player, some developments are anticipated over his future.
The interest in the goalkeeper comes after an excellent 2017-18 season, in which he established himself as the starting stopper at the Stadio Olimpico.
Alisson not only earned a reputation as one of the best goalkeepers in Serie A, but he also cemented his place in the Brazil XI ahead of the FIFA World Cup, where the Selecao were knocked out by Belgium in the quarter-finals.
As things stand, Liverpool arguably have the greatest need for a goalkeeper, as they have Loris Karius as their starter.
The German had a disastrous UEFA Champions League final, as two errors cost his team dearly. In pre-season Karius has continued to look shaky, as a handling mistake allowed Tranmere Rovers to score in a recent friendly encounter.
Given there were doubts about his pedigree prior to those errors against Real Madrid in May, plenty who attend Anfield would be disappointed to see Karius as the side's starting goalkeeper come the first day of the campaign.
As noted by Carl Markham of the Press Association, the supporters haven't forgotten about the mistakes he's made in recent weeks:
With Simon Mignolet and Danny Ward the other goalkeeper options available to manager Jurgen Klopp, if Liverpool want a significant upgrade they will need to go out and sign a new stopper.
Alisson would be an expensive purchase but an exciting one. The Brazilian is not only nimble, sharp and instinctive in his goalkeeping, his distribution from the base of the side sets him apart from many others in European football.
Having already landed Naby Keita, Fabinho and Xherdan Shaqiri, the Reds have spent huge money this summer in an attempt to challenge for the Premier League and Champions League prizes. If he comes in, Alisson might just be the most important acquisition of the lot.