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British media reported the deal will cost the Premier League champions around £20 million ($NZ38 million) with £5 million in add-ons.
Thiago will leave the German giants a year before his contract is set to expire and is expected to sign a four-year deal with the Anfield club.
"Thiago was an exceptional player for seven years for Bayern Munich," Bayern coach Hansi Flick said.
"The way he played he gave many options to the team, he was a surprising player because he solved problems in ways people did not expect.
"It was a joy working with him. He is an absolute professional and a great person. I congratulate [Liverpool coach Juergen] Klopp. He gets a top player and a sensational person."
Rummenigge had earlier confirmed the move, saying the two clubs had reached an agreement and the player was leaving Munich.
Having come through Barcelona's youth system and made 100 appearances for the senior side, Thiago moved to Bayern in 2013 when the German club appointed Pep Guardiola, becoming a vital cog in central midfield.
Thiago has won the Bundesliga title with Bayern every season since his move to Munich and also lifted the Champions League trophy last season.