The Merseysiders are understood to be keen on a new goalkeeper either now or during the summer, with both Simon Mignolet and Lorius Karius having failed to cement the No. 1 jersey at Anfield.
Despite having been given numerous chances by Klopp, the duo have largely failed to impress, turning in a number of error-strewn displays.
Liverpool's search for a new stopper is believed to have led them to Alisson, though his current employers are reluctant to see him leave.
Italian source 'Il Tempo' reports that Roma have placed a £44m price tag on the 25-year-old's head, with an opening offer of significantly below that amound rejected out of hand.
The Brazilian has kept 12 clean sheets in 27 games so far this season, including three in six games in the Champions League, catching the eye of a number of top sides.
Speaking to 'Sky Italia' recently, Alisson was quizzed about interest from the Premier League and he refused to rule out a switch to the English top flight, saying: "I am happy in Rome. It's always a pleasure to feel that lots of people are monitoring you but my head is focused on Roma. I leave my future in the hands of God.
"In June, maybe we will speak about it again but for now I am concentrated on Roma.
"Together with my team-mates, we have done a great job to not allow the loss of [Wojciech] Szczesny to be felt by the fans.
"I have worked very hard and want to continue along that track."
Roma already face a battle to keep hold of two of their other stars in the form of defender Emerson Palmieri and striker Edin Dzeko, with Premier League champions Chelsea circling the pair.