"Today morning, he had an accident on the Nuptse wall and died. It seems he slipped," Ang Tsering Sherpa, head of the Nepal Mountaineering Association, told AFP.
Steck, 40, was on Everest to acclimatise before attempting in May to summit the world's tallest peak followed by neighbouring Lhotse, connecting a series of ridges to design a never before climbed route.
Everest and neighbouring peak Nuptse share a common ridge, which is where Steck slipped and fell, according to a government official.
"He skidded off about 1,000 metres from (Mt Nuptse) camp two early morning on Sunday. Other climbers ascending Everest saw him and asked for his rescue," said Dinesh Bhattarai, director general at the Department of Tourism.
Steck was climbing alone when he died. His partner, Tenji Sherpa, had sustained severe frostbite earlier and was recovering at a lower camp. His body was recovered by Nepali guides and flown by helicopter to Kathmandu.