The boat was found capsized earlier this morning, after it lost radio contact late on Saturday, a coast guard official in the southern island of Jeju said by telephone.
The bodies of 10 people were recovered in the waters near the island of Chuja, which lies between the mainland south coast and Jeju, the official said.
Three people were pulled from the water and airlifted to hospitals, the coast guard official said, adding they were expected to survive.
"There were six people without life jackets including the captain hanging onto the capsized boat," one of the survivors said in an interview with Yonhap. "One by one, those who lost strength slipped away."
Another survivor was quoted as saying that around five others failed to escape the boat when it turned over. However, the coast guard official said no one was found trapped in the boat during their search for more survivors.
Around 21 people are expected to have been on the boat, a spokesman from the Korea Coast Guard said during a briefing.
Most of those on board were on a fishing expedition to Chuja, a popular fishing area, a second coast guard official said. Some of them were from an online fishing club based in Busan.
President Park Geun-hye has called on the rescue and recovery services to do their utmost in the search for the missing, her office said in a statement.
In April 2014, a passenger ferry, Sewol, sank off the southwest coast killing about 300 people, most of them children on a school outing, triggering a national outrage over what was seen as an ineffective rescue operation.
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