The blaze started shortly after 02:00 local time (10:00 GMT Thursday) at the No 14 hospital in Ramenskiy village.
Most of the victims are believed to be local patients. There are fears that the death toll could rise further.
The cause of the blaze is being investigated, with one report saying it was caused by a short circuit.
The fire broke out in part of the building where patients with severe conditions were being held, local officials were quoted as saying by Russian news agencies.
Firefighters later extinguished the blaze and have so far found 12 bodies.
A local official was quoted as saying that all the windows had metal bars, and most of the victims were found in their beds.
A nurse managed to lead to safety two patients.
It is believed that 41 people were in the hospital when the fire began.