At least five of the 42-man crew died and 20 were rescued after the deep-sea trawler went down about 2,000km (1,250 miles) south of New Zealand.
Rescuers have scaled back the search, saying it was unlikely that any of those missing would survive for long.
The crew included nationals from South Korea, China, Indonesia, the Philippines and Russia.
South Korea's foreign ministry said the 614-tonne ship, the Number One In Sung, sank at around 0430 South Korean time on Monday (1930 GMT on Sunday).
Twenty sailors were rescued by another fishing boat operating nearby, the South Korean foreign ministry said in a statement.
New Zealand rescuers said it was not clear why the boat sank. Conditions were calm and no SOS was sent.
Survival times in the water in the area would be about 10 minutes without life-jackets or immersion suits, rescuers said after scaling back the search.
"It (is) becoming increasingly unlikely further survivors (will) be found," rescue controller Dave Wilson told AFP news agency.
Ross Henderson, a spokesperson for Maritime New Zealand, earlier said all vessels in the remote area had been told to go to the site to help search for survivors after the incident was reported.
A helicopter was also sent to the area, he said.