Karius was signed by Klopp from his former club Mainz for £4.7 million and the assumption was he would become the new No. 1.
However, a broken hand in pre-season delayed his availability but after a comfortable midweek run-out against Derby in the EFL Cup he was handed the chance to stake his claim.
But despite that vote of confidence the 23-year-old knows there is still plenty of work to be done.
“Obviously [me and Mignolet] train really well because we know we have a strong opponent so I think it is positive for us and the club,” he said.
“We know we have to be at our best to play otherwise there will be another strong keeper waiting.
“It is really healthy competition and we don’t have anything against each other and we try to help each other in training and get on okay so there is no issue between us.
“The one who doesn’t play is going to be frustrated but the manager has to pick 11 players.”
On arriving from the Bundlesliga, Karius asked for—and was given—the vacant No. 1 shirt but he insists that was not a conscious decision to set out his ambitions early.
In fact, he played down the significance of his choice of shirt.
Karius added: “The No. 1 is just a shirt really. It doesn’t really say much but it was free so why not take it?
“If you have the No. 1 shirt and you don’t play any games it doesn’t really help you.
“I think for the people outside it has a bigger meaning than it actually is.
“It is nice to have this number but it does not automatically mean you are in that position.”
Karius’ Premier League debut may not have finished with a clean sheet but he was rarely troubled by a Hull side who were reduced to 10 men early on and never posed much of a threat.
Instead it was an easy afternoon for the German, although he insists he did not stand by and revel in some of the thrilling football being played by his outfield team-mates.
“I had games like this before but I will probably have them more here,” he said.
“I hope it is like this as that means we are playing really well but as a goalkeeper you have to stay in the game because it goes really fast and you only need one shot on goal [to concede].
“It doesn’t really change much whether you have a lot to do or not much.
“When we score a nice goal I am happy for the other guys and then I am a fan but the only time I can really have a break is when we score or when there is an injury break, otherwise you are focused.”