The current Reds number one made two costly errors as Klopp's charges lost 3-1 to Real Madrid in May.
Although Karius was later adjudged to be suffering from concussion, Souness believes that those errors would come back to haunt the German had he started 2018-19 as first-choice and made another mistake.
Roma and Brazil custodian Alisson Becker is believed to be on the verge of a switch to Anfield, which would push Karius down the pecking order and perhaps out of the club, and the Scot believes that Klopp is doing the right thing.
"I think as much as Jurgen would have wanted to be loyal and do right by the goalkeeper, he's got to go out and get another one," Souness told Sky Sports News.
"That's simply because if Karius was to make the same mistake, or a mistake in a big game, it'd be down to the manager not correcting that situation. It's obvious to everyone - look at what happened in the Champions League final.
"So if Klopp is out to get Alisson or another top goalkeeper, he's doing the right thing."