Authorities had lost track of the boat from the South Korean port city of Busan in the early hours, the official said.
The vessel had been carrying 27 people and foreign nationals were among those rescued, said the official, adding that two South Korean nationals had died.
The South Korean coast guard had received a distress call at 4:33 a.m. (1933 GMT on Thursday) located 24 kilometers (15 miles) away from an island off the coast of Jeju, the Yonhap news agency reported.
President Yoon Suk Yeol has ordered all available resources to be deployed for a search and rescue operation, his office said in a statement.