Kagawa rose to prominence under Klopp after joining Borussia Dortmund from Cerezo Osaka, of the Japanese second division, in 2010.
The 26-year-old says he remembers "every moment" with Klopp fondly, with Kagawa making an emotional departure to Manchester United after winning the Bundesliga twice.
"I remember him expressing his emotions out all the time, his goal celebrations, encouraging the team, attitudes to communicate and give love to each player," Kagawa told Omnisport.
"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, the goal in the derby match, the game I scored to seal the title and the celebrations.
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"The trust between us got bigger and bigger and then I could play in his team again. [The farewell] is a bit exaggerated, but I went to see him to say thank you for the two years in the last moment as my move was almost decided.
"When I went see him and did the last greetings, the tears naturally came out. 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.
"Showing my further development and better performance is the repayment to him from me. My career still goes on and I do not want to be remembered as I was good only during that period. And now I have got the new manager, so I want to achieve good results."
Kagawa has been impressed with the start Klopp has made at Liverpool and is interested to see how quickly the Reds can fully adapt to his methods.
"I am interested in how much and how quickly Klopp, the manager that I know, can change the team at Liverpool," the Japan midfielder said.
"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 drastically changed now 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."