According to initial police reports, an 85-year-old man apparently killed his 81-year-old wife, then shot himself.
A police murder response team is under way to the Upper Austrian town of Braunau am Inn in order to investigate the incident.
The alarm was raised by a family member, who discovered the bodies after he had tried unsuccessfully to reach the pair by telephone.
The man was in possession of the handgun legally.
The town, which is close to the German border, has been the scene for a dispute for the past several years over the ownership of the house in which Hitler was born.
Old people, and especially older men, have one of the highest rates of suicide. In particular, feelings of isolation and becoming a burden to others are major risk factors for such suicides.
A similar case occurred nearby in early January, when human remains were found dumped in three suitcases in Lake Traun in Western Austria. A man's body was found a short time later. Both were in their 70s.
Another case took place in September in Vienna's Hietzing district, where a 77-year-old man is believed to have killed himself as well as his 76-year-old wife.
Earlier the same month, bodies of a mother and daughter were found in Vienna's Simmering district, after an apparent murder-suicide.
In many cases, firearms were involved. There has been a trend recently for increasing private ownership of firearms, as well as a proposal to allow police officers to be issued weapons for private use.