Steck, 40, died at Camp One of Mount Nuptse, in the Himalayas, and it appears the 40-year-old had been planning to climb Everest and nearby Mount Lhotse next month.
Alan Hinkes - the first and only Briton to have climbed all 14 of the world’s mountains over 8,000m - said: “Even the armchair mountaineers in Britain will be gutted.
“Everybody knows of him. He was the man of the moment. He was on everybody’s lips. He was the Wayne Rooney or David Bekcham of the mountaineering world.
“He’ll be greatly missed, there’s no doubt about it. He’ll go down as one of the greatest ever.”
Mr Hinke said Steck - known as the Swiss Machine - was famous for doing technically difficult climbs very fast and solo - including classic Alpine routes like the north face of the Eiger, which he completed in two hours and 47 minutes without using a rope.
He said: “He was pushing the envelope, but he was at the top of his game.
“He was only 40. Some younger people might think that that’s old but, for a mountaineer, he was right at the peak of his performance.
“He would certainly have gone on to even greater things.”
Mr Hinkes said he believed Steck was planning something “mega” on Everest.
But he said he knew what he was doing was extremely dangerous.
Mr Hinkes said: “He was pushing the envelope risk-wise but he knew it. He was very, very experienced and technically proficient, but unfortunately something’s happened.
“He was aware of the risk. He’d been on camera saying he knew this could happen to him.
“There but for the grace of God.”
Mr Hinkes said he bumped into Steck many times around the world.
“He was a great bloke,” he said. “He wasn’t bombastic. He was unassuming, really
“He was wiry, like a coiled spring. But he always had a glint in his eye. You could see he had a sense of humour and he had a wry smile. He always seemed happy and he seemed chilled.”
Expeditions organisers said Steck’s body has been recovered from the site and taken to Lukla, where the only airport in the Mount Everest area is located.
He was the first casualty in the spring mountaineering season in Nepal that began in March and will end in May.
Hundreds of foreign climbers are on the mountains to attempt to scale Himalayan peaks in May when there are a few windows of favourable weather.
In 2013, Steck completed the first solo climb of the Annapurna south face in Nepal after almost losing his life in a fall there in 2007, earning him the Piolet d’Or - considered the Oscar of mountaineering.
In 2015, Steck decided to climb all 82 peaks in the Alps higher than 4,000 metres (13,100ft) - travelling between mountains by foot, bike and paraglider only.
He completed the feat in 62 days, helping cement his reputation as the “Swiss Machine”.
Steck said in an interview last month with Swiss Tages-Anzeiger newspaper that he considered himself an “outsider” in the mountaineering scene because athletic achievement was more important to him than adventure.
Asked about his forthcoming Everest-Lhotse expedition, involving a quick climb from one peak to the other including an overnight in the “death zone”, Steck said: “When I’m on Everest I can stop at any point.
“The risk is therefore quite small.
“For me it’s primarily a physical project. Either I get through, or I don’t have the strength for the whole traversal.”
Asked what he would consider to be success on his expedition, Steck told Tages-Anzeiger: “Of course I want to climb Everest and Lhotse. But that’s a very high goal.
“Failure for me would be to die and not come home.”