After being used mostly in cup competitions for the first part of the season, the 24-year-old has now started the Reds' last four Premier League games in a row to displace Simon Mignolet as first pick.
Klopp has been tipped to make an overhaul of his goalkeeping department this summer, but Karius has said that he is determined to prove his worth before then.
"I just want to present myself well and then I'll have the chance to play at this club for the next season as well. That's the challenge facing me personally," he is quoted as saying by Sky Sports News.
"I also want to help this team achieve as much as we can. As well as the top four, we still have the Champions League. We want to go forward in that and why not win it?
"If you read the press as a Liverpool player then every week we would have five new signings in every position. You can't go crazy about the speculation.
"When I look at the training, I think we do have good quality already with the keepers. I think we've been a bit unlucky at times.
"Often you don't have much to do. Then if you don't do something 100% right it weighs a lot heavier than for other goalies at smaller clubs who have several more saves to make in a game. That's life at a big club like Liverpool and you have to deal with it."
Karius has been at Anfield since 2016 and has three-and-a-half years remaining on his current deal.