But Bild have reported that an opportunity has now opened for the Reds to swoop in for the Austrian defender.
Christian Falk posted on Twitter that the third round of negotiations between Alaba and Bayern have “failed”.
The 28-year-old is out of contract next year, meaning he could leave at that point or sign a pre-contract in January.
And Liverpool could enter the fray as Football Insider have reported that the defender is on their target list.
While the Reds can boast previous business with the German side, having signed Thiago Alcantara over the summer.
Alaba has spent his entire senior career at the Allianz Stadium, winning nine Bundesliga honours and two Champions League honours in nearly 400 appearances.
He has only missed three games this season – and his seven appearances have all come in the centre-back position.
His potential availability in the coming months will come as good news to the Reds after distancing themselves from Schalke youngster Ozan Kabak.
And the latest on Fabinho suggests they will need to act sooner rather than later in recruitment.
“Losing another centre-back was exactly the last thing we needed,” Klopp said.
“[Fabinho] felt his hamstring and that’s not good.
“He didn’t feel it too much: he said he could have played on but not make any more sprints.”