"The incident happened two days ago in Ebonyi state. The couple were French tourists and they were camped in the bush when they were attacked," national police spokesman Emmanuel Ojukwu told AFP.
"The man was killed while his wife was injured and some valuables, including cash, were taken away from them. We suspect a robbery attack."
Ojukwu said some arrests had been made.
Local media said the incident occurred on Tuesday at the town of Abaomege, where the couple had pitched their tent to pass the night before heading to the southeastern port city of Calabar.
A spokesman for the French Embassy in Abuja, Georges Vanin, confirmed the incident and described the crime as "purely financially motivated".
"It was a retired couple who were heading from Paris to South Africa by car via the African coast and they had to cross Nigeria," he said by telephone from the capital.
"They decided to stop on the way to Calabar... They were clearly spotted when they were putting up their tent."
Vanin said the couple were familiar with Africa and the man who died was retired from the army while his wife was a former nurse, without revealing the couple's identities or ages.