Klopp was speaking to Gary Lineker on The Premier League Show, a weekly magazine programme which airs for the first time on BBC Two at 22:00 BST on Thursday.
Shankly won the First Division three times, the FA Cup twice and the Uefa Cup once while managing Liverpool between 1959 and 1974, while Paisley won three European Cups, six First Division titles, the Uefa Cup and three League Cups from 1976 to 1983.
Klopp ‘wants to drive on big bus through Liverpool’
“I would like to celebrate something each season over the next six years,” he added. “Not the small thing, really celebrate something – driving on the big bus through Liverpool. That would be nice.
“I’ve said it before, but it is not important what people think when you come, it is important what they think when you leave. That is when you need to be judged.”
Klopp moved to Liverpool after seven years at Borussia Dortmund.
The German club are famous for their ‘Yellow Wall’, a stand packed with 25,000 fans that creates one of the most intimidating and impressive atmospheres in world football.
However, Klopp says the atmosphere generated during Liverpool’s dramatic comeback at Anfield against Dortmund in the Europa League quarter-finals in April eclipsed anything he had experienced before.
He said: “When I was at Mainz there were 20,000 people going nuts, then I went to Dortmund and you would get 80,000 – not too bad – but then the last half-hour against Dortmund was the best I’ve ever had.
“I felt no-one could stop us.”
‘People should be excited to watch us’
In The Premier League Show, Klopp also gives an insight into his managerial philosophy.
He explains how he deals with leaving players out of his matchday squads, and also said he could understand why people might think he “is crazy” with his emotional antics on the touchline.
Klopp also said his number one aim as a manager is to excite fans.
“Having memorable games in a row, that is my first target,” he said.
“People being unable to wait for the next game, that is what football should be. If you can do this very often then you will be successful.”
Klopp keen for winter break
Having experienced the benefits of a winter break in Germany, Klopp is keen to see it introduced in the Premier League.
He highlighted England’s performance at Euro 2016 in France as a reason why it would be beneficial for players.
England were knocked out at the last-16 stage by Iceland, and Klopp said: “They played good football but they didn’t score enough.
“I know a lot of these players, they have a good attitude and want to work. Dele Alli, what a season, but then maybe it felt like 500 games in his legs – he was not 100%. Harry Kane – what a season, and then he cannot show it at the European Championship.
“You need fresh legs.”
Watch The Premier League Show online and on BBC Two at 22:00 BST on Thursday.