"Both men died while descending from the summit on Monday," an official with the tourism ministry told AFP from Everest Base Camp.
Sung Ho-Seo, 34, of South Korea was attempting the climb without supplementary oxygen and died on his way down the mountain.
Mohammed Hossain, 35, from Bangladesh, died in his tent a few hours after successfully climbing the summit.
"The exact cause of death is unknown, but altitude played a part," said the official, Gyanendra Shrestha, adding that the bodies would not be recovered until after the summit season ended so as not to interrupt other climbers.
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Five other climbers have died on the 8,848-metre (29,029-foot) mountain this season.
Early in the season Mingmar Sherpa, 47, a member of an elite team known as the "icefall doctors" who set up climbing routes, plunged to his death.
DaRita Sherpa, 47, died from what is believed to have been cardiac arrest earlier this month. Commercial guide Lobsang Sherpa, 22, also plunged to his death. A 50-year-old Russian climber, Alex Bolotov, was found dead near the famed Khumbu Icefall crevasse on May 15.
Some 300 people have perished trying to reach the summit during the last six decades. The bodies of some of them remain on the mountain.
May is considered the best time for climbing in the Nepalese Himalayas because of mild weather and some 300 people have reached the top of Everest so far this year.
But a brawl that erupted last month between three European climbers and Nepalese guides on the mountain cast a shadow over this year's season, which marks the 60th anniversary of the maiden ascent by Tenzing Norgay and Edmund Hillary.