The initial fee is £20m but add-ons could take this up to a total of £27m. Those add-ons are based on appearances and Liverpool winning trophies.
Thiago will sign a four-year contract and wear the No 6 jersey at Anfield.
It had initially looked like any deal would be dependent on the future of Georginio Wijnaldum, who is also in the final year of his Anfield contract, but Liverpool have decided to make their move now.
Thiago helped Bayern win the treble last season, as they beat PSG in the Champions League final last month in Lisbon.
However, in July Bayern chairman Karl-Heinz Rummenigge revealed that Thiago had told the club that he wanted a new challenge.
Rummenigge admitted that talks between the player and sporting director Hasan Salihamidzic over a new deal had broken down as he wished to leave Germany and said the club would accept a “reasonable price” for him.
The Spanish international started his career at Barcelona and was handed his debut for the Catalan side by Pep Guardiola in 2009 at the age of 18.
He was an unused substitute as Barca beat Manchester United in the Champions League final the following season.
Thiago joined up with Guardiola again in 2013 when the new Bayern boss took him to Bavaria where he went on to win seven Bundesliga titles and four DFB-Pokal cups.
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