The Reds are stepping up their efforts to land the Spain international, with the fee potentially rising to £27m (€30m/$35m) in add-ons.
After months of speculation official contact has been made between the two clubs, with Liverpool now set to agree terms with the player, who is keen to move to Anfield.
A four-year deal has been agreed, with Thiago set to wear the No.6 shirt vacated by Dejan Lovren at Anfield. He will, however, not be in contention to face Chelsea at Stamford Bridge on Sunday, but could make his debut against Lincoln in the Carabao Cup next Wednesday.
Bayern CEO Karl-Heinz Rummenigge confirmed the agreement on Thursday, telling reporters: "I can confirm that Bayern has finally reached an agreement with Liverpool. It was Thiago's great wish to do something new again at the end of his career."
Sources have told Goal that the Reds' move for Thiago is not dependant on any outgoings, despite speculation linking Gini Wijnaldum with a move to Ronald Koeman's Barcelona. Young striker Rhian Brewster has also been tipped for a switch away, with Sheffield United keen on signing the England under-21 star.
Thiago's arrival, though, will provide a huge boost to the Premier League champions, whose sole business so far this summer has been the £10.8m signing of left-back Kostas Tsimikas from Olympiacos.
With Thiago having told Bayern that he wished to move on and seek a new challenge , Liverpool have always been at the front of the queue to sign the 29-year-old, despite reports suggesting Manchester United and Barcelona were also interested.
Klopp has remained tight-lipped on the reports, but dropped a hint last week when refusing to dampen the speculation at a press conference at Melwood.
"If I could invent a word now that would end speculation, I could become really rich!" the Reds boss told reporters.
"I can't end speculation as long as the transfer window is open, that's all I can say. Did I mention that Thiago Alcantara is a really good player? Long ago!
"It's nice that we are linked with him but that's pretty much all, really. He is a good player, Liverpool are a big club, that's nice, but apart from that, really nothing else to say on it. Unfortunately, maybe good for you [the media], I cannot end the speculation."
The move for Thiago represents a significant change in transfer policy for Reds owners Fenway Sports Group, who have generally targeted younger signings during their time on Merseyside. The likes of Mohamed Salah, Sadio Mane, Roberto Firmino, Fabinho, Naby Keita and Alisson Becker all arrived for big fees, but in their early-to-mid 20s.
But Thiago, at 29, represents a proven world-class talent whom Klopp believes will add a new dimension to his squad as they look to defend their league crown, as well as competing for the Champions League.
The fact that he had entered the final year of his contract at Bayern, and that he had expressed his desire to move to Anfield, meant the decision was taken to pursue a deal.
Liverpool have had concerns over the financial impact of Covid-19, which led to them pulling out of a move to sign Timo Werner from RB Leipzig earlier in the summer, but believe a £27m move for Thiago was simply too good to turn down.
They will, however, still seek to offload a number of fringe players. Marko Grujic, Harry Wilson and Xherdan Shaqiri will all be sold if a suitable offer is received, while Loris Karius will also be allowed to leave. Brewster, 21, is attracting interest across the Premier League, though no final decision has been taken on his future at this stage.