"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 to summit the world's tallest peak in May, using a never before climbed route.
Everest and neighbouring peak Nuptse share a common ridge, which is where Steck apparently 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.
The accomplished alpinist sought to pioneer new routes throughout his mountaineering career, earning the nickname "the Swiss Machine" for his solo record ascents in the Alps.
Steck made global headlines in 2013 when he and two other Western climbers traded blows with a group of furious Nepali guides over a climbing dispute on Mount Everest.
The brawl shocked the mountaineering community, causing a damaging rift between Western climbers and the often lowly-paid Nepali guides who are essential for commercial expeditions to the crowded summit.
An angry Steck swore never to return to Everest, telling a Swiss website that his "trust (was) gone".
But he was back in the Himalayas only months later, this time to scale Mount Annapurna, the world's tenth highest peak, via its steep Southface wall.
After 28 hours of climbing -- all without bottled oxygen -- and a dangerous brush with an avalanche, Steck became the first mountaineer to complete a solo ascent of the 8,091-metre peak.
Controversially, he offered no photographic proof of his accomplishment, saying the avalanche knocked his camera out of his hand.
Nevertheless, he was awarded the Piolet d'Or, mountaineering's top accolade, for the 2013 climb.
Born in the town of Langnau im Emmental, near the Swiss capital Bern in October 1976, Steck was a devoted climber by the age of 12.
As an 18-year-old, he climbed the north face of the Eiger, an achievement that attracted attention and later sponsors, setting him on a course to become a professional climber.
He soon became one of the most prominent names in mountaineering after scaling some of the world's most daunting peaks, often alone and without basic safety equipment such as fixed ropes or bottled oxygen.
But, in an interview with AFP nearly two years before his death, Steck said he was never inspired by a quest for fame or "records".
"Personal pleasure alone dictates my approach", he said in an August 2015 interview in the French Alps town of Sigoyer.
He summarised his approach to climbing as a blend of "preparation, instinct and emotion".
He told AFP that he never wanted climbing to become a "business" and was content when his income exceeded what he previously made as a carpenter.
Steck was however attacked by some in the mountaineering world for resisting the use of GPS to track his movements or refusing to record photographic evidence of his achievements, criticism that he dismissed as "jealousy".