The talented young goalkeeper was loaned to Mainz from Manchester City in 2011, with the deal made permanent in January 2012.
He made his first-team debut on December 2012 but failed to make a significant breakthrough until the 2013/14 season, in which he would make 23 appearances across all competitions.
Karius' stock exploded from there, firmly setting himself as Mainz's first-choice goalkeeper over the 2014/15 and 2015/16 seasons.
The Premier League club signed Karius, then 22, in the summer of 2016 for £4.75m on a five-year deal.
The German displaced Simon Mignolet as Klopp's first-choice goalkeeper during the 2017/18 season, but two errors in the 2018 Champions League final loss to Real Madrid called into Karius' quality and maturity.
"He got a chance in the first-team and did very, very well. And then when you have an offer from Liverpool, you have to do this. And why not?"
Given the success the Reds have achieved with the Brazilian as their goalkeeper, Muller says Karius can't have too many complaints.
"I think after the final they bought Alisson, then it is very difficult for Loris to play," the Western Sydney Wanderers forward added.
"But in the end Liverpool made the right decision because they have one of the best goalkeepers in the world and he saved a lot of games."