Usukhjargal Tsedendamba, 53, and Purevsuren Lkhagvajav, 31, were last in contact Sunday evening from Camp 4, which is situated less than a kilometre (3,300 feet) below the summit.
Tsedendamba's body was found around an altitude of 8,600 metres (28,215 feet) on Friday morning after days of search and rescue operations that had been hampered by bad weather.
"It seems he was returning from the summit. Search is continuing for the other climber," Lakpa Sherpa of 8k Expeditions, which organised the duo's climbing permits, told AFP.
Sherpa said the two men were climbing without guides and that their walkie-talkie had been found in their tent.
Nepal's tourism department said in a statement on Tuesday that another team had spotted the pair "heading towards Everest's summit" on Monday morning.
Hundreds of climbers have flocked to Nepal -- home to eight of the world's 14 highest peaks -- for summits in the spring climbing season when temperatures are warmer and winds are typically calm.
Nepal has issued more than 900 permits for its mountains this year, including 419 for Everest, earning more than $5 million in royalties.
Around 80 climbers have already reached the 8,849-metre (29,032-foot) summit of Everest after a rope-fixing team reached the peak last month.
Two climbers have died on Makalu, the world's fifth-highest peak, this year.
French climber Johnny Saliba, 60, died at an altitude of 8,120 metres (26,640 feet) during his summit push on Sunday.
A 53-year-old Nepali guide died on the same peak last week while descending from the summit.