The German was at fault for two goals as Real MadridÂ won 3-1 in KievÂ to lift their third consecutive Champions League.
And according to the Daily Mirror, the Reds will now intensify their search for a new No 1 at Anfield.
They say Romaâ€™s Alisson and Jan Oblak of Atletico Madrid top Jurgen Kloppâ€™s wishlist this summer.
The Mirror add that Klopp wanted to give Karius a chance to impress, however, his performance in Kiev ended his chances of being Liverpoolâ€™s long-term No 1.
Liverpool have been monitoring Alisson all season, while the Reds would have to break the bank for Oblak.
The Slovenia international has an ÂŁ80m release clause in his current Atletico deal.
The Mirror also say that Cologne No 1 Timo Horn and Southamptonâ€™s Alex McCarthy could also be potential alternatives for the Reds this summer.
Simon Mignolet is also set to leave Anfield this summer, especially if Klopp lands a new goalkeeper in the coming weeks.
On his horror show in the Champions League final, 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. Itâ€™s 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