It said that the bodies belong to a man and woman, both Greek Cypriots. Authorities are examining all possibilities.
Police and forensic pathologists are at the scene which has been cordoned off.
Nicosia CID officers are taking statements from residents of the apartment block and relatives of the deceased.
In statements at the scene, deputy police chief Demetris Demetriou said the bodies of a woman and man were found at around 9.30 pm.