The accident occurred as the Jeju Air plane, carrying 175 passengers and six crew on a flight from Thailand, was landing at the airport in the south of the country, the report said.
Two people were found alive as the rescue mission was continuing, according to Yonhap.
Authorities were working to rescue people in the tail section, an airport official told Reuters.
Photos shared by local media showed smoke and flame engulfing parts of the plane.
Acting President Choi Sung-mok, who was named interim leader of the country on Friday after the previous acting president was impeached amid an ongoing political crisis, ordered all-out rescue efforts, his office said.
A Jeju Air spokesperson said the airline was checking the reports.