Forty-four-year-old British climber Robin Fisher reached the summit on Saturday morning but collapsed on the return, just 150 metres back down the slope.
On the northern Tibetan side of the mountain, a 56-year-old Irishman died on Friday morning, without reaching the summit. He died in his tent at the North Col pass at 7,000 metres.
Four climbers from India and one each from the United States, Austria and Nepal have already died on Everest in the past week. Another Irish mountaineer is missing presumed dead after he slipped and fell close to the summit.
A traffic jam of climbers in the Everest "death zone" has been blamed for at least four of the fatalities, heightening concerns that the drive for profits is outweighing safety.
Nepal issued a record 381 permits to mainly foreign climbers, costing $11,000 each, for the spring season.
Every climber with a permit must be assisted by at least one local guide, adding to the summit logjam.
With the short window of suitable weather due to close soon, bottlenecks of scores of climbers wanting to achieve the ultimate mountaineering accolade have built up each day.
An estimated 600 people had reached the summit via the Nepal side by Friday. At least 140 others have been granted permits to scale Everest from the northern flank in Tibet.