But hopes of landing Roma's Alisson Becker have so far been dashed by the Serie A giant's high valuation of the Brazil global.
Karius, though, is the man in the spotlight, and given the high-profile nature of his errors in Kiev, that will remain the case for some time yet - something Klopp is well aware of.
"Loris concedes that goal, but we can not start a story always after each mistake".
The German added that Karius will continue to be criticised until "he has a few fantastic games".
"We can not change that".
Zlatko Dalic says Croatia's World Cup progress is no surprise
With Modric and Rakitic controlling the midfield, Croatia have had more possession than their opposition in four of five games at the World Cup.
His performance has inevitably led to speculation that Liverpool FC could be set to look to bring in goalkeeping reinforcements this summer as they look to challenge for the title next season.
And Klopp replied: "You can all stop that immediately, promise?"
"Two brilliant footballers make this mistake, but we don't talk about it".
It appears that one of the Tranmere players wasn't too impressed with Karius' howler, and he showed him little sympathy.
And the Liverpool manager maintained his support for the under-fire stopper after his mistake in the game with the newly-promoted League Two side.
Liverpool are next in action on Saturday when they travel to Bury.