The Blues confirmed the injury, saying it would not be known how long Davies will be sidelined for until after surgery.
"Bradley will have an operation early next week and an accurate timescale on his recovery will be assessed after the operation," the club said in a statement.
Interim Wales head coach Rob Howley must hope it will be a brief spell on the sidelines as he was already facing a lengthy injury list, with Davies his last fully-fit lock until this latest setback.
On Friday, centre Ashley Beck suffered an ankle ligament problem of his own, while Rhys Priestland (Achilles tendon), lock Luke Charteris (knee) and prop Aaron Jarvis (knee) have all been ruled out of the tournament.
Ospreys second-row forward Alun-Wyn Jones, meanwhile, is currently recovering after shoulder surgery and flanker Dan Lydiate has not yet played since suffering a serious ankle injury earlier this season.
Jones and Lydiate are both given chances of featuring at some stage of the Six Nations, while Ian Evans has made encouraging progress in his return from a knee injury and hopes to be available.
Wales kick off their campaign in three weeks against Ireland.
Davies suffered the injury despite sitting out the Blues' Heineken Cup trip to Toulon this weekend as his partner Laura had gone into labour, and he had some good news too on Saturday morning with the birth of their first daughter, Elle Cheryl Davies.