Wales and Lions forward Hibbard has been ruled out of the Christmas fixtures against the Scarlets and Cardiff Blues after chipping a bone in his wrist in the Heineken Cup win over Castres.
But Bishop is set to be sidelined for at least four months after undergoing surgery on a back problem.
Ospreys head physio Chris Towers said: "Andrew, has undergone surgery in an attempt to correct the same long-standing back problem that ruled him out of action last season.
"It is difficult to predict timescales for a return to action for him, but we are looking at a best-case scenario of around four months.
"Richard chipped a bone in his wrist on the weekend against Castres which is likely to rule him out for the next couple of weeks and the away games against Cardiff and Scarlets.
"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 on January 3."