Karius was at fault for two of Real Madrid's three goals when they beat Liverpool 3-1 in May's Champions League final, after which it was revealed the 25-year-old German was suffering from concussion.
Liverpool have been strongly linked with the 25-year-old in recent months but appeared to shelve their interest after being put off by Roma's hefty valuation and there was talk that Klopp was ready to keep faith with Loris Karius.
Alisson is the number one choice for the Brazil National team and played in every game for Brazil at the World Cup in Russian Federation.
The Mirror are reporting that Roma are looking for somewhere in the region of £66 million for the goalkeeper as they feel that Liverpool shortchanged them last summer in their purchase of Mo Salah.
Chelsea are also interested in the Brazilian number one, but Liverpool will hope their long-term pursuit gives them the advantage.
But Liverpool are now in pole-position to sign Alisson, with Chelsea yet to make an improved offer for him.
Klopp was already considering a new goalkeeper before that poor performance.
Liverpool are wary that Real Madrid are closing in on Chelsea keeper Thibaut Courtois which could mean the Londoners enter the race for Alisson.
The potential bid would shatter the world record fee for a goalkeeper, which was the £35m Manchester City paid Benfica for Ederson last summer.