The 30-year-old will have a scan and may need surgery.
"It's not looking too great...you try to look at best case scenarios, six to eight weeks but worst case scenario they've got to operate," Ospreys defence coach Brad Davis told reporters.
"I know players before who have had bicep tears and they've had to have operations and they could be (out) 12 to 16 weeks, possibly longer."
Wales were already without flanker Sam Warburton for the Feb. 3-March 17 tournament after the British and Irish Lions captain underwent knee surgery in December.
Number eight Taulupe Faletau is also doubtful for the start due to a knee ligament injury. Wales kick off their campaign against Scotland in Cardiff on Feb. 3.
(Reporting by Alan Baldwin, editing by Ed Osmond)
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