The 25-year-old is joined by prop Gethin Jenkins - who will miss the whole Six Nations campaign - and scrum-half Richie Rees on the sidelines.
Ospreys physio Chris Towers said: "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.
"In light of the location of the injury, we are anticipating that he could be out of action for up to eight weeks."
Bishops absence from the squad raises the prospect of centre Gavin Henson making his comeback to the national set up ahead of this autumn's World Cup.