The Daesh-controlled Amaq news agency said that the attack was carried out by two militants of the group in response to France's role in the US-led anti-terrorist coalition.
According to the France Info news outlet, both attackers were killed by police.
The BFMTV broadcaster later reported that the dead hostage was a priest.
French President Francois Hollande has met with the family of the deceased and expressed his support for all Catholics.
"Catholics have been attacked, all Catholics, but also all the French," Hollande said, adding the nation must stand united.
Earlier, Pope Francis has expressed his horror at the 'barbaric killing' of the priest during the incident.
"The pope… shares the pain and horror of this absurd violence," Vatican spokesman Federico Lombardi said as quoted by AFP, adding that the attack created "immense pain and worry".
Daesh is notorious for its human rights atrocities and brutal tactics, such as public beheadings and the stoning of women.
