Al-Shebab, the Somalian armed Islamist group, say they have executed a French intelligence agent who they had held captive since 2009.
The al-Qaeda linked group said they killed Denis Allex on Wednesday.
French officials dispute the claim, saying they believe Allex was killed soon after a failed rescue attempt on Saturday.
A senior al-Shebab official told the AFP news agency on Thursday that "audio and video [of the execution] are available and will be released any time we decide".
He added that the hostage was killed in Bulomarer, a rebel-controlled town south of the Somalian capital, Mogadishu.
Sheikh Abdiasis Abu Musab, the spokesman for military operation for al-Shebab, said: "Let Muslims enjoy his execution and the French cry."
French troops launched an unsuccessful raid on the town to free Allex on Saturday, which French officials say left two French soldiers and 17 rebel fighters dead.
Al-Shebab had sentenced the hostage to death earlier on Wednesday, saying in a statement it had "reached a unanimous decision to execute the French intelligence officer".
The group also cited "France's increasing persecution of Muslims around the world, its opressive anti-Islam policies at home, French military operations in ... Afghanistan and, most recently in Mali."
'Manipulating the media'
The French army accused al-Shebab of "manipulating the media" on Wednesday.
A video showing Allex appealing to the French president Francois Hollande was released by the group in July.
"Mr President, I am still alive, but for how long? That depends upon you, for if you do not reach an agreement for my release, then I am afraid this will be the last message you receive from me," said Allex reading from a sheet of paper.
"My life depends on you."
Allex, believed to be a psuedonym for the agent, was kidnapped in Mogadishu in July 2009 along with another French intelligence officer, Marc Aubriere from the Directorate-General for External Security.
Aubriere escaped captivity a month later.