The national emergency management agency said up to 50cm of rain has fallen since Friday in the province of Kangwon in the mountainous east of the country and that more was expected.
An orange alert has been issued for the capital, Seoul, and its surrounding regions and for Kangwon. The orange alert puts residents of affected areas on standby for possible swift evacuation and advises children and the elderly to remain indoors.
The alert is a step below the category-red warning that shuts down schools and workplaces and bans access to motorways.
The emergency management agency said there was a "high likelihood of large-scale disasters" from the heavy rain.
Mudslides killed 11 people in Kangwon and another 18 people were missing. Two people were also found dead in other areas of the country.
More than 200 people remained isolated as roads were inaccessible because of flooding.
Roads been blocked by mudslides
Electricity and telephone communications were cut off in many parts of the region.
Torrential rains also flooded about 1,300 homes across the nation, forcing nearly 3,000 people to seek refuge elsewhere, the agency said.
More to come
In Seoul, which received about 30cm of rain, a flood advisory was issued for the Han River, according to the weather agency.
The Korea meteorological administration said it expected more heavy rainfall nationwide until Thursday as a seasonal rain front moved southward.
The country's president, Roh Moo-hyun, visited the emergency agency on Sunday and called for "through preparations to minimise damage".
North Korea, which borders Kangwon, reported on Sunday that heavy rains had "seriously flooded farmland" across the country, but did not mention any deaths.