Real Madrid are also said to be pursuing the 25-year-old, who has established himself as one of Europe's leading goalkeepers since taking over from Wojciech Szczesny this season.
The German keeper broke down in tears after the game and has since apologised to the Liverpool supporters over social media.
He added: "Liverpool can win the Champions League next year but need two more players to add to the squad". When we goalkeepers see our colleagues make mistakes, we suffer with them.
Real Madrid's Sergio Ramos celebrates after winning the Champions League Final soccer match between Real Madrid and Liverpool at the Olimpiyskiy Stadium in Kiev, Ukraine, Sunday, May 27, 2018.
Roma have slapped a £79 million price tag on their key man in the hope that it will prevent the Merseysiders from making a move for the player.
"But life goes on".
Roma sporting director Monchi has said the club have not received an offer for goalkeeper Alisson amid continuing speculation linking him with Liverpool.
"The difference is that you need a little bit of luck in a football game, especially a final, and we not only had no luck - we had none plus bad luck".
However, Germany head coach Joachim Loew, now in northern Italy for a World Cup training camp, sent Karius, who has yet to play an worldwide, a text message of support following Saturday's final.
"I would like to meet him so as to tell him that he only needs courage, or maybe common sense, so as to understand that the best lessons are the hardest".