Klopp won both his and Liverpool’s first trophies of note since 2012 with a 2-0 victory over Tottenham in the Champions League final.
The German was appointed Reds manager in October 2015 with the club tenth in the Premier League and in the Europa League.
Within three full seasons he has taken them to their sixth European crown and first in 14 years, and Carragher feels that Klopp is now indispensable.
The club legend would even prefer to lose Van Dijk, who he considers the league’s best defender, over the manager.
“I think he is the No 1 at the club. What I mean by that is he is the most important figure,” he told Sky Sports.
“Sometimes in the team you have a star player but if you’re asking me now if I’d rather lose Van Dijk or the manager, I’d rather lose Van Dijk.
“I think he is a special manager. I don’t know what his contract situation is but I’d give him a new one.
“Hopefully now he can push on. He has got that monkey off his back if you like – the first big trophy, first time he’s won the Champions League, all these finals without winning and all this talk.
“Hopefully for him and those players it isn’t a one-off like it was for us in 2005.
“We won the FA Cup the year after but we didn’t really kick on and win any more trophies. Hopefully as the manager he can do that.”