Usukhjargal Tsedendamba, 53, and Purevsuren Lkhagvajav, 31, were last in contact Sunday evening from Camp 4, which is situated less than a kilometer (0.6 miles) below the summit.
Tsedendamba's body was found around an altitude of 8,600 meters (28,215 feet) on Friday morning after days of search and rescue operations that had been hampered by bad weather.
"The second body was also found on Friday, but needed verification. We confirmed it was him yesterday," Pemba Sherpa of 8k Expeditions, which organized the duo's climbing permits and base camp stay, told AFP.
"We are trying to bring the bodies down."
Four guides had been sent for their search and rescue.
Lkhagvajav's body was found near an area called the balcony, a small platform at an elevation of around 8,400 meters (27,560 feet).
Sherpa said that 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-meter (29,032-foot) summit of Everest after a rope-fixing team reached the peak last month.
Two climbers have died on nearby Makalu, the world's fifth-highest peak, this year.
French climber Johnny Saliba, 60, died at an altitude of 8,120 meters (26,640 feet) during his summit push last week.
A 53-year-old Nepali guide died on the same peak last week while descending from the summit.