The Premier League club will pay £20m across Thiago's four-year contract, with the potential for £5m in add-ons, and the midfielder will wear the No 6 jersey.
Thiago flew in to Manchester from Munich on Friday morning to complete a medical and finalise a deal which will make him Liverpool's second summer signing.
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 Paris Saint-Germain in the Champions League final last month in Lisbon.
With the move to Merseyside imminent, Thiago took to social media to post a goodbye video which said: "I have made the most difficult decision of my sports career.
"I will close this chapter in this wonderful football club where I have grown as a player for seven years.
"Triumphs, joys and also hard times but what I am most proud of is that I arrived in Munich as a young man full of dreams and now I will leave completely fulfilled with the club, a history and a culture.
"Here I learned to love a tradition that I am now saying goodbye to but which I will never forget.
"My decision is purely a sporting nature as a soccer player, I want new challenges to develop myself further. Bavaria will always be my home, thank you FC Bayern."
Bayern boss Hansi Flick also paid tribute to Thiago on Thursday and revealed he had said his goodbyes to players and staff at the club.
"Thiago is an exceptional player who spent seven hugely successful years here," Flick said. "He's a wonderful person and an outstanding professional. It was very emotional today when he said goodbye to us all."
In July Bayern chairman Karl-Heinz Rummenigge revealed Thiago had told the club he wanted a new challenge.
Rummenigge admitted 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 Spain 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.
Robertson: Thiago a world-class player
Liverpool defender Andy Robertson said "until something is signed on the dotted line" he is not paying too much attention to the expected move of Thiago, but said the Spaniard is "full of talent" and would be a welcome addition to any team.
"I've watched him, he's a fantastic player - a player full of talent," Robertson told Sky Sports News.
"He would add to the quality of a squad in probably any team in the world, so if we get him, great, if we don't then it is what it is. We'll wait and see until it's definitely confirmed.
"[Bayern Munich's] quality is there to be seen; we watched their Champions League final when we were away at pre-season, and they've got world-class players in every position pretty much, and he's definitely one of them.
"Look, [they are] a team full of quality and he's played at the highest level so fair play if we are linked with him."
Analysis: Thiago signing shows how high Liverpool are regarded
Sky Sports News report Kaveh Solhekol told The Transfer Show:
"Liverpool signing Thiago shows how high Liverpool's standing is right now in world football because this is a player who is the best in the world at what he does. He's been at Bayern Munich for seven seasons, he's won a title every season, he's just won the Champions League and was one of the best players in the competition last season.
"Bayern were aware his contract was running out next summer. They put a new contract in front of him, a four-year deal and they thought it was all agreed, he was going to sign on the dotted line but at the last minute, he had a change of heart and decided to leave this summer. Bayern decided to let him go and he wants a new challenge which is at Liverpool.
"From Liverpool's point of view, Bayern were initially asking for €30m and there was no way Liverpool were going to pay that. The way they've structured this deal is that they are paying £20m but over four seasons so £5m per year for Thiago. When you look at it like that, it is the best lease purchase deal I have ever seen to get a player that good for just £5m per year.
"That's why a lot of Liverpool fans on social media are saying their sporting director Michael Edwards is a genius."
'Thiago deal a no-brainer for Liverpool'
Sky Sports News reporter Vinny O'Connor:
"Liverpool's stance throughout the summer was that they would not be signing Thiago Alcantara - that changed a little a couple of weeks ago when the suggestion was they would only try to sign him if someone was leaving but they did not believe that was going to happen.
"Now Liverpool have decided to go ahead regardless of that and they think Thiago is the man for them.
"Jurgen Klopp had also been consistent throughout the summer - Liverpool would not be doing business unless a really good opportunity presented itself and a deal was there to be done.
"This is a deal that Liverpool could not turn down.
"At 29 Thiago does not necessarily fit the FSG model of buying players young, coaching them and increasing their ability and value, but the out-and-out quality of Thiago, combined with his experience of performing on the biggest stage meant this particular move was a no-brainer.
"He is a player for the here and now, untested in the Premier League but someone who is used to performing on the biggest stage for Barcelona, Bayern and Spain.
"The signing should also help to solve an issue that Liverpool have recognised recently - Trent Alexander Arnold alluded to it in an interview a couple of weeks ago - the game plan needs to change.
"It has been in the back of Liverpool's minds to change the dynamic in midfield. It is not a case of things going stale but they are keen to give the opposition something else to think about now. Liverpool have been playing the same way for a couple of years and Thiago brings them something completely different.
"The signing of Bayern Munich's treble-winning midfielder renews that competition for places and says to the other players in that position they have to raise their game again.
"Klopp knew Leeds would present a real challenge and something different on the opening day of the Premier League season and the signing of this type of midfielder allows Liverpool to change things a little themselves and catch other teams out.
"They have now decided to move from a position of strength to freshen things up and give other teams something new to think about."
Carra: Thiago brings something different
Sky Sports' Jamie Carragher:
"Liverpool have got five or six midfield players - they rotate well because they need energy in that position - but a lot of the players are pretty similar.
"Thiago would be something different - less in energy but more technical - and I think he would be a really good addition.
"Liverpool are world class at the top end of the pitch, world class in terms of their defence and goalkeeper; I think they're top class in midfield.
"Something could be added technically and Thiago fits the bill, so fingers crossed he comes."
Analysis: How Thiago could transform Liverpool
Sky Sports' Adam Bate:
The commentator was just saying how comfortable Paris Saint-Germain were looking when it happened. "They are quite happy for Bayern to have the ball in their half," he said. But it was at that precise moment that Thiago Alcantara weaved his magic, firing a crisp pass between the lines.
The ball was in the back of the net seconds later. The goal was enough to win the 2020 Champions League for Bayern Munich and while scorer Kingsley Coman was the official man of the match, the role of Thiago in the team's triumph escaped absolutely no one.
At 29 and reluctant to sign a new contract, Bayern realised they might have to sell. Liverpool are now ready to pounce and it is easy to see why the move might appeal. Jurgen Klopp's side can do almost anything. But they cannot do what Thiago can do…
Source: Sky Sports