Al-Qaeda in the Islamic Maghreb (AQIM) told ANI that it killed Philippe Verdon on 10 March, in retaliation for France’s intervention in Mali.
Mr Verdon and another Frenchman were seized in the town of Hombori.
French troops were sent to Mali in January after al-Qaeda-linked militants threatened to march on the capital.
In recent weeks French-led forces have been fighting militants in the remote Ifoghas mountains of northern Mali.
Early on Wednesday the French foreign ministry said it was trying to verify the report by Agence Nouakchott d’Information.
Philippe Verdon and Serge Lazarevic were said to be on a business trip when they were captured at their hotel in November 2011.
AQIM claimed it was behind the kidnapping and published photos of the pair in captivity.
Besides Mr Verdon, a total of 14 French nationals are being held by Islamist groups in Africa.
Six of them are thought to be detained in Mali. They include four hostages kidnapped by AQIM at a uranium mine in northern Niger in 2010.
France currently has 4,000 troops in Mali, who are assisting thousands of Malian, Chadian and other African troops.
The French government intervened when a coalition of Islamist and separatist rebels, which had seized the north of the country last year, threatened to move on the capital Bamako.
The Islamist groups had taken over major cities, including Gao, Kidal and Timbuktu, and imposed strict Islamic law in the aftermath of a coup in March 2012.
Since the intervention began, this territory has been recaptured but fighting is still continuing in desert mountains.
France plans to withdraw its troops from Mali next month, with West African countries expected to take over in the run-up to elections due in July.