A South Korean fishing boat carrying nine crew, including seven Indonesians, capsized off the country's southern coast on Saturday, leaving six missing, local authorities said.

The 29-ton vessel overturned in waters 68 kilometres (42 miles) south of an island in the coastal city of Tongyeong on early Saturday morning, according to South Korea's Yonhap news agency.

So far three crew members have been rescued but all were unconscious, the Korea Coast Guard's Tongyeong office told AFP.

"We are still looking for six missing individuals," official Lee Ho-jun said.

South Korean President Yoon Suk Yeol ordered relevant authorities to "do their best to save lives by mobilising all available personnel and equipment, including navy and fishing boats," his office said in a statement.

The boat that capsized had set sail from the country's southernmost island of Jeju on Thursday morning, and was engaged in fishing, Yonhap said.

The news agency said patrol boats, navy vessels, and aircraft have been deployed for the ongoing search efforts.