"It is still unknown what has occurred but it is believed that they have perished on the mountain. The family have been informed of the news."
On Christmas Day, Dr Bishop, a father of two, decided to attempt the summit that had eluded him twice before.
After hearing he had not returned, his family hoped he and his German companions had found shelter.
But after yesterday's fruitless search, they were bracing for the worst.
"I've been talking to my mother," said Dr Bishop's eldest son, Joshua. "Everyone is still having faith and hoping for a good outcome ... but we're just coming to terms with the fact there's been a tragedy and probably all three of them have lost their lives."
Police say the search will now be re-evaluated. Further aerial searches will take place over the next few days.
The Christchurch Alpine Cliff Rescue team have left and the Department of Conservation rescue team are back on standby at Mt Cook village.
Mike Bishop, who started climbing as a medical student about 30 years ago, had not planned to climb with the Viellehners but after they met, they decided to attempt the summit together.
Joshua Bishop, 27, said his father "wasn't a Mt Everest, alpha climber" but a highly experienced climber who loved the thrill of reaching a summit. "He would get to the top of a mountain and be cheering and really excited. He found it quite euphoric. It wasn't a junkie rush - he just loved being up there and close to his God."
The Bishop family are staying home, praying and supporting one another.
Joshua Bishop said they would want to return his father's body to Australia.
The Bishops are also thinking about the Viellehner family who have lost a father and son.
"We're thinking so much about that family.
"We're just a few hours' flight away but they're all the way back in Germany," he said.
"It's such a sad tragedy."
He praised the search and rescue efforts, which made his family feel as if they were "looking for the President [of the United States]".
"I've never experienced this type of tragedy before, but to know people are over there looking for your dad, updating you on what they are doing and asking what morethey can do, it's wonderful to have."