The Japan international played under Klopp for two spells at Dortmund before the German quit his job at the club last year.
The duo won two Bundesliga titles during their time together and Kagawa believes he will be a great asset at Anfield.
"I am interested in how much and how quickly Klopp, the manager that I know, can change the team at Liverpool," Kagawa told Omnisport.
"He is a great coach and the results prove it, so I hope it will continue for him and I am interested in how much they can contend as a team this season.
"Already they have drastically changed and I have read articles that they won eight games in the last 10 games [seven in 11]. I do watch the games sometimes when I have time or when they broadcast on TV."
Kagawa added: "Every moment with him is a good memory for me rather than picking a particular one up. It is also good memory to win the titles with him.
"He trusted and featured me. I was not famous coming from the second division of the J.League, so I totally appreciate that. It is great memories with him.
"He helped me to build up the steps leading to Manchester United, the club that everyone all over the world knows, the club that I have known since a little kid, in only two years after I joined from Cerezo."