Bayern's sporting director Karl-Heinz Rummenigge said they're willing to let him go for a decent offer - which is thought to be around £30m and talks are said to have begun with the Premier League champions.
Liverpool are not due to be big spenders and that amount for a 29-year-old, who only has a year left on his deal, may not have been seen as good business by Anfield transfer guru Michael Edwards, but they'd be getting a proven operator at a world class level.
Thiago has been billed as just the player Klopp needs in that midfield - a more technically gifted player than they have at the moment in their three-man engine room - that currently provides pressing, pace and power but not too much guile and creativity.
The question is, where and how will Thiago fit in?
He's not an athletic powerhouse runner, so fitting Thiago into the Liverpool midfield would mean tweaking Klopp's 4-3-3 system somewhat.
Up to now, the Reds haven't really needed to mess about with formations too much, but a condensed season next year and a smaller squad means they may well have to. Liverpool's midfield trio have been all about hunting and chasing down the opposition, limiting space and time, winning the ball quickly and moving it on even quicker in transition to their dangerous front three.
Thiago offers something different, something more like Philippe Coutinho - a playmaker and dribbler - something that hasn't been seen at Anfield since the Brazilian left. He's been linked with a return but there's no sign that he'd be welcomed back any time soon.
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