One fisherman had been found unconscious and not breathing in waters 7.4 kilometers (4.6 miles) south of the boat, and rescue workers were airlifting him to a hospital on the nearby island of Jeju.
The coast guard and navy were deploying boats, helicopters and a patrol plane to search the waters near Jeju island for survivors, said Lee Geun-han, an official from the coast guard in Jeju.
Rescue efforts, which also involved civilian fishing vessels, were being slowed by strong winds and large waves in the area.
Lee said the coast guard received a report about the fire at around 7 a.m. A court guard helicopter found it about an hour later, and the upper side of the boat had been burned completely.
A photo released by the coast guard showed the 29-ton boat engulfed in black smoke.
Six of the boat’s crew were South Koreans and the other six were Vietnamese, Lee said.
They departed from the mainland port of Tongyeong on Nov. 8 to catch hairtail and had planned to return to the port on Monday.