The Spain international is on the verge of joining the Reds from the Bundesliga champions as he eyed a new challenge.
He had spent seven trophy-laden years in Bavaria – with the most recent season ending in a treble including the Champions League.
But with his contract expiring next summer, Thiago told Bayern chiefs he would not sign a new deal.
That forced them into looking for suitors, while the midfielder’s mind was already made up.
Reds fans are over the moon to see a player of such class arrive while Liverpool legend Carragher has outlined what the playmaker will give to the club.
He said: “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.
Liverpool are keen to repeat last season’s success – and you don’t want to miss a thing, surely?
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Liverpool got their title defence off to a shaky start as they edged out a seven-goal thriller against Leeds.
In terms of transfers. The Reds have agreed a deal for Bayern Munich’s Thiago Alcantara, but Georginio Wijnaldum continues to be linked with Barcelona. There’s a lot to keep track of.
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“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.”
Speaking to Sport1, Hoeness said: “It is a difficult situation. A while ago we agreed on a new four-year contract. Some days later he said: ‘I’m looking for a new challenge.’ Which is totally okay.
“I think he obviously has an agreement with Liverpool or Manchester United or both of them. Now both clubs are bluffing, and no one has contacted Bayern Munich officially.
“From my point of view, that has no style. They’re trying to blackmail us, by waiting until the last week, when they’re going to make a cheap offer.
“In that case FC Bayern have to ask themselves the question: do they want to go along with that or do they tell the player that he has to stay.
“It’s going to be a hot three weeks until [the transfer window closes on] October 5.”