The 26-year-old lasted just 23 minutes of the derby victory and Ospreys coach Sean Holley admits it is not looking good for the front row.
"He is a tough cookie so it's quite serious, early indications aren't great," said Holley.
The same injury meant Hibbard was unable to be in contention as Wales were hit by a hooker crisis in the RBS 6 Nations.
Hibbard won his four Wales caps in 2007 but was hoping that a grandstand end to the season would force his way into Warren Gatland's summer plans.
South Africa visit Cardiff on June 5 before a two-Test tour of New Zealand.