The bodies were discovered on the farm in the village of Nouvion et Catillon in the Aisne department at 8.30am on Tuesday morning by firefighters.
According to reports in France the victims were two adults, aged in their 50s and their three children, two boys and girl aged between 14 and 20.
The alarm was raised by an employee at the farm who discovered the bodies.
Investigators say they are keeping an open mind about what led to the deaths but they believe a “family tragedy” (drame familial) is the likely cause.
A source told Le Parisien newspaper that it is likely the father killed his wife and his three children before turning the gun on himself.
“We don't know what provoked the tragedy, we haven't found a suicide letter,” the source added.
Detectives in the nearby town of Laon are in charge of the investigation.