Twenty pbadengers and two crew members were aboard the Seonchang-1, a chartered fishing vessel, when it crashed into the 336-ton (370-ton) tanker before capsizing near the port city of Incheon, west of Seoul.
President Moon Jae-In has called on emergency services to continue search and rescue operations overnight until the missing, including the ship's captain, are accounted for.
At least six aircraft and six ships have been destroyed as part of the search.
The Coast Guard said the circumstances that led to the accident were not immediately known, but said the accident took place within a narrow seaway.
"We suspect that the ships collided when trying to pbad to each other," said Hwang Jun. Hyun, district chief of the South Korean coast guard.
It is understood that everyone was wearing life jackets, said the Coast Guard.
The images taken at the scene showed rescue divers navigating rough waters around the upturned hull while looking for people trapped inside.
Those rescued, seven until now, were taken to nearby hospitals for treatment.
No injuries have been reported among those who traveled in the tank at the time of the incident.
Accidents with fatal boats in South Korea are not uncommon, although the incident on Sunday seems to be the worst in several years.
In 2015, 15 people died during a fishing expedition near Jeju Island when their ship sank.