The woman was strangled while the man was discovered under a carpet, the website of Le Matin newspaper reported, citing unidentified sources.
No information was available about how the man died, the website said.
Geneva cantonal police said they found the bodies after breaking into an apartment at 2 Rue Jean-Simonet in the Châtelaine neighbourhood on Saturday around noon.
They had been contacted earlier by relatives who were concerned because they had not heard from the couple, originally from the canton of Schwyz.
Police said the investigation is being conducted by investigators from the criminal brigade and by forensic specialists in a case that appears to be criminal matter.
The investigation is being directed by prosecutor Linda Chabal.
Police declined to provide information about the case.