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Police said he was Jonas Legaard Sorensen (21) who was visiting New Zealand from Denmark.
The man was last seen nearly a week ago.
Sergeant Mark Kirkwood said West Coast Police Search and Rescue, along with Land Search and Rescue, searched the Cassel Flat area yesterday morning.
Police said yesterday afternoon that, while the body had not yet been formally identified, it was believed to be the man searchers had been looking for since he was reported overdue on Monday.
Sgt Kirkwood said the 21-year-old went into the area by himself on June 3 to hunt for tahr and was last seen in the Cassel Flat hut on June 6, by other hunters.
Police believe he had a personal locator beacon, but it had not been activated.
The area concerned is hazardous and recent weather has been cold and wet. A heavy rain warning was in force for the area yesterday.
The man's family in Denmark have been advised.
The death will be referred to the coroner.