Nepalese media reported Hossain, 35, died in his tent on Monday, several hours after successfully climbing the summit.
His sponsor a Bangladeshi private television channel Desh TV confirmed the death.
Its head of marketing Sadequr Rahman Chowdhury told bdnews24.com the television channel was the media partner and sponsor of Hossain’s endeavour.
Hossain, member of Bangla Mountaineering and Tracking Club (BMTC), was popularly known as Sajal Khaled.
South Korean climber Sung Ho-Seo, 34, also died while attempting a climb without supplementary oxygen.
"The exact cause of [their] death is unknown, but altitude played a part," local official Gyanendra Shrestha was quoted as saying by the media.
He added the bodies would not be recovered until after the summit season ended so as not to interrupt other climbers.
Both men perished in the "death zone" -- above 8,000 meters, notorious for its difficult terrain and thin air.
Five other climbers have died on the 8,848-meter (29,029-foot) mountain this season.