Police were called shortly after 7 a.m. on Tuesday following a report of a man showing no signs of life near a balcony at a residential building on Jagiełły Street in the city's Pod Dalnią neighbourhood.
After entering the flat, officers found three more bodies: a woman and two young men.
An aggressive dog inside the property had to be safely captured by police and animal shelter staff before officers could enter.
Daniel Prokopowicz, a spokesman for the District Prosecutor’s Office in Kielce, said preliminary findings suggested that the deceased were members of one family: a 55-year-old man, a 52-year-old woman and their two children, aged 18 and 16.
Their identities were still being verified, he added.
Two prosecutors, a forensic doctor and police officers were examining the scene.
Investigators were collecting evidence to establish what happened and determine the circumstances of the deaths.
Prokopowicz said the investigation was ongoing and that the authorities had not ruled out different scenarios.
The precise sequence of events and the cause of death remained to be established.
(pg/gs)