At least 177 patients - majority elderly - were in the building when the fire broke out, according to Song.
The government of Asia's fourth-largest economy, with one of the world's fastest ageing populations, has faced criticism in recent years over poor safety standards, including the Sewol ferry disaster of 2014 in which more than 300 people, mostly schoolchildren, drowned.
Fire in a South Korean hospital, has led to the death of 37 people, and injuring 130 more.
The death toll rose rapidly throughout the morning as some of those initially pulled from the blaze succumbed to their injuries.
Dozens of fire trucks and two helicopters were sent to the hospital as thick smoke blanketed nearby streets. The blaze was was extinguished after one hour and 40 minutes, Yonhap said.
Most of the victims were on the first and second floors of the hospital's six-story general ward, where its emergency room and intensive-care unit were located. It started around 7.30 a.m.in an emergency room of the hospital, Xinhua news agency reported. The National Fire Agency said officials were trying to identify other victims. The nursing home and hospital between them have about 200 beds.
Details of the fire including its cause were unknown.
It is South Korea's deadliest fire in nearly a decade and the death toll is expected to rise with several of the injured in critical condition.Firefighters said the victims appeared to have died from smoke inhalation.
President Moon Jae-in convened an emergency meeting with senior advisers and expressed regret over the blaze. The hospital has been operating since 2008. He ordered officials to supports to those rescued, find the exact cause of the fire and work out measures to prevent future fires, said his spokesman Park Su-hyun.
Firefighters expect the number of casualties will rise further, according to South Korea's official news agency, Yonhap.
"The nurses were coughing heavily and their hands were bloody ... it looks like they were injured while helping people evacuate", she said, adding one male medical worker also rushed into the smoke-filled hospital with a fire extinguisher in a desperate bid to put out the raging fire.
The latest blaze occurred not long after a conflagration at a fitness centre in Jecheon city killed 29 people last month.