Jurgen Klopp wants to install the Brazil international as his new No 1 after Loris Karius and Simon Mignolet failed to convince in the role.
And The Times report Liverpool chiefs are following his wishes by pushing ahead with a move to take Alisson to Anfield.
Liverpool are said to be close to sealing the world-record signing of the 25-year-old having improved on the bid of £62m (€70m) they made on Tuesday.
Respected Merseyside-based journalist Paul Joyce says Liverpool’s new offer of £66.8m (€75m) has been accepted by Roma.
Liverpool and Roma were apparently in talks on Wednesday morning over how payments for Alisson will be structured.
And the Reds are now making plans for the stopper to have a medical, with permission granted to talk personal terms.
Chelsea have also been credited with interest in Alisson in case they lose contract rebel Thibaut Courtois this summer.
But The Guardian claim the Blues have been left standing in Liverpool’s wake and are yet to make a bid of their own.
Liverpool’s progress towards a deal for Alisson has meant the former Internacional ace spoke to Klopp on Tuesday.
And he is believed to have his heart set on a move to last season’s beaten Champions League finalists.
Real Madrid could enter the fray if Courtois reneges on his decision to head back to Spain with them.
But they will have to get involved quickly with Alisson seemingly destined to join his former Roma team-mate Mohamed Salah at Liverpool.
Alisson will continue Liverpool’s summer spending which has already seen Fabinho, Naby Keita and Xherdan Shaqiri signed by Klopp.
And he might end up being followed to Anfield by Lyon captain Nabil Fekir, with Liverpool still keen on the French playmaker.
