Concise News reports that Klopp claimed his first trophy as Liverpool boss as they beat Tottenham 2-0 to win the Champions League in Madrid on Saturday night.
It was also the first European silverware of the German’s career and ended a torrid run of six consecutive cup final defeats dating back to 2012.
Speaking on Sky Sports, Carragher says of the manager: “I think he is the No 1 at the club. What I mean by that is he is the most important figure.
“Sometimes in the team, you have a star player but if you’re asking me now if I’d rather lose [Virgil] 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, the 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.”
When told that Klopp’s current contract runs until 2022, Carragher joked that it should be extended by ’17 years’.
Klopp signed that deal back in the summer of 2016 after guiding Liverpool to the finals of the League Cup and the Europa League in his debut campaign at Anfield.
He has since led them to back-to-back Champions League finals and they also pushed Manchester City to the wire in the Premier League title race.