The 30-year-old is set to miss the PRO12 games against Cardiff Blues and Scarlets over the festive period but he is expected to return next month.
Hibbard started the decisive final test of the Lions' historic 2-1 series victory in Australia over the summer, having appeared off the bench in the opening two tests.
The hooker will be an instrumental part of Lions coach Warren Gatland's plans with Wales during the Six Nations, for which he is expected to return in plenty of time.
"[The injury] is likely to rule him out [of] the away games against Cardiff and Scarlets," said Ospreys physio Chris Towers.
"We expect this to settle fairly quickly though and remain optimistic about his prospects of being available for the game against the Scarlets at the Liberty Stadium."