The Reds have been eyeing a new goalkeeper all season, with Jurgen Klopp having issues with finding a solid No 1.
Loris Karius, who was at fault for two goals against Real Madrid in the Champions League final on Saturday night, has been first choice in recent weeks.
Simon Mignolet has also been in and out of Kloppâ€™s starting line-up this year due to inconsistent form.
And according to Palmeri, Liverpool are desperate to land Romaâ€™s Alisson, with the transfer expert claiming a ÂŁ61m bid will be enough to land the Brazil international.
He tweeted: â€śLiverpool had seen rejected back in February a â‚¬50 millions (ÂŁ44m) offer for Romaâ€™s Brazil international Alisson.
â€śRoma having as minimum bid a â‚¬70m (ÂŁ61m) price tag.â€ť
On his horror show in the Champions League final, where Real won 3-1, Karius wrote on Instagram: â€śI havenâ€™t really slept until now.
â€śI know I messed it up with the two mistakes and let you all down The scenes are still running through my head again and again.
â€śIâ€™m infinitely sorry to my team-mates, for you fans, and for all the staff.
â€śIâ€™d just like to turn back the time but thatâ€™s not possible.
â€śIt’sÂ even worse as we all felt that we could have beaten Real Madrid and we were in the game for a long time.
â€śThank you to our unbelievable fans who came to Kiev and held my back, even after the game.
â€śI donâ€™t take that for granted and once again it showed me what a big family we are. Thank you and we will come back stronger.â€ť