Johann Viellehner, 58, and his 27-year-old son Raphael, together with Australian Michael Bishop, 53, were last seen roped together as they climbed towards the summit of the mountain on Monday.
"There is no sign of them. The situation is grim. It was hoped that if they had survived the storm they would have been spotted from the air today. The family have been informed of the news," police said in a statement.
"It is still unknown what has occurred to the men but it is believed that they have perished on the mountain."
More than 30 centimetres of snow has fallen on Mount Cook over the past few days.
The two older men were described as experienced mountaineers and Bishop has previously climbed in the Mount Cook region on New Zealand's South Island.
Mount Cook, also known by its Maori name Aoraki, is New Zealand's highest peak at 3,754 metres, and is a popular challenge for mountaineers from around the world because of the skills required to reach the summit.
It has claimed more than 200 lives since it was first climbed in 1894.