The Italian club, which is also fending off interest from Chelsea for their Brazilian number one, have indicated they want £66 million.
Negotiations continue, but the mood has shifted on Merseyside and in Rome over the last 24 hours as the clubs edge closer to a deal.
Chelsea are also interested in the 25-year-old, who may eventually have his choice of Premier League clubs. Liverpool are aware of the Londoners’ efforts but hope their long-term interest in the keeper gives them the upper hand.
Having suggested Alisson was not for sale at any price earlier in the transfer window, Roma’s position has shifted significantly.