Bishop who has been capped 15 times by Wales, played a part in all four of the recent Autumn Internationals but is likely to miss the first four games of this year's tournament
"Andrew picked up an ankle injury against the Blues on New Year's Eve, and we have now been able to establish the full extent of the ligament damage sustained through an MRI scan," said Ospreys head physio Chris Towers.
"In light of the location of the injury, we are anticipating that he could be out of action for up to eight weeks."
Bishop's absence will also be a big blow to the Ospreys as they enter a fortnight of Heineken Cup action.
They are currently third in Pool 3, but just three points separate the top three sides and current leaders Toulon are still to visit the Liberty Stadium.