An alpine rescue team managed a brief flight over the area in a light plane early in the morning but saw no sign of the three before being forced back by bad weather.
"Conditions were quite rough for flying, and the weather forecast is for deteriorating weather," police officer Brent Swanson said.
Australian Michael Bishop, 53, Johann Viellehner, 58, from Germany and his son Raphael Viellehner, 27, were last seen roped together as they climbed towards the summit of the mountain on Monday morning.
Alpine guides said conditions were reasonably good, but crevasses were an issue along with unstable snow bridges.
The two older men were described as experienced mountaineers and Bishop has previously climbed in the Mount Cook region on South Island.
Mount Cook, also known by its Maori name Aoraki, is New Zealand's highest peak at 3,754 metres (12,388 feet), 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 conquered in 1894.