The Spanish midfielder completed a £25million move to the Premier League champions earlier this week after seven successful years for the Bundesliga giants.
Rummenigge said: "After the transfer was finalised, I saw Thiago in the parking spot waiting for me.
"He hugged me for five minutes, cried and thanked me for making it possible for him."
Thiago left Bayern Munich as a Champions League winner after he helped the German giants complete a sensational treble under head coach Hansi Flick.
The 29-year-old has also won the last seven Bundesliga titles with Bayern while he has also lifted four German Cups and the 2013 Club World Cup trophy.