The Seoul-based Yonhap news agency said Song had collapsed due to altitude sickness and fallen off a cliff, quoting a diplomat at the South Korean embassy in Kathmandu.
It said the climbers were part of a team of old classmates from the same high school in the central city of Daejeon.
About a dozen members flew to Nepal at the end of March to mark their school's 50th anniversary by climbing the peak. They were due to return home later this month.
Tilak Pandey, a tourism ministry official at Everest base camp, said separately that a 33-year-old Nepali-born Canadian woman named Shriya Shah had also been killed on Sunday.
Shah was born in Kathmandu and grew up in Mumbai, India, according to her website. She lived in Toronto and described herself as "an entrepreneur, political activist, social worker, and above all, a daring lady."
Speaking of her Everest expedition, she wrote: "This is my dream and passion, and want to do something for my country. Nothing is impossible in this world."
Another ministry official, Deependra Paudel, said 55-year-old Chinese climber Ha Wenyi had also been found dead, at 8,600 meters.