The pair and the manager will leave this summer.
Klopp said: "In all the years that I have been involved in football, I am not sure I have come across too many players who are more loved than Joel Matip. I'm not even sure it would be possible to say anything bad about him.
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"A wonderful professional, a wonderful footballer and a wonderful human being we have been blessed to have him with us for as long as we have and now all we can do is wish him well as he heads off in a new direction."
On Thiago's departure, Klopp added: "Before he even came to Liverpool I believed that if you really love football it would make a lot of sense if you watched Thiago Alcantara play.
"Technically, he is so, so good, a talent who could play in any team in the world, and it was a privilege to have him with us."