The scene first aired last year and was repeated on a DVD release in March. But in a particularly bad piece of timing for HBO, stories about it spread online this week, when the network premiered a documentary on Bush's father, former President George H.W. Bush.
The former president's features on the prosthetic head were partially obscured by long, scraggly hair and the picture flashes by quickly.
The producers said they often order prosthetic body parts in bulk for "Game of Thrones" because it can be too expensive to make new ones.
"We meant no disrespect to the former president and apologize if anything we said or did suggested otherwise," the producers said in a statement.