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 northern town of Hombori.
France sent troops to Mali in January saying al-Qaeda-linked militants could over-run the capital, Bamako.
In recent weeks French-led forces have been fighting militants in the remote Ifoghas mountains of northern Mali.'Second-tier extremists'
Early on Wednesday the French foreign ministry said it was trying to verify the report by Agence Nouakchott d'Information (ANI).
ANI quoted a man who identified himself as Ghairawani, a spokesman for AQIM, as saying that Mr Verdon had been executed.
AQIM at a glance
Formed in 2007, mainly by Algerian fighters
Operates across the Sahara Desert
Fought in Mali alongside local militants
Thought to have between 600 and 800 fighters
Chad says its troops killed its top commander Abdelhamid Abou Zeid in Mali in March
Pascal Lupart, who runs a support group for some of families of the French hostages, said they feared that the hostages had been handed over to "second-tier" extremists after the French campaign in Mali intensified, Associated Press news agency reports.
The hostage's father, Jean-Pierre Verdon, told AP the ANI report was "not a confirmation'' of his son's death, and the family was awaiting French government comment.
Philippe Verdon and Serge Lazarevic were said to be on a business trip when they were captured at their hotel in Hombori, about 160km (100 miles) from the regional capital Gao, in November 2011.
French troops captured the town from militant Islamists about two weeks after they intervened in Mali.
Besides Mr Verdon, a total of 14 French nationals are being held by Islamist groups in Africa.
France plans to withdraw its 4,000 troops from Mali in AprilSix of them are thought to be detained in Mali. They include four hostages kidnapped by AQIM at a uranium mine - vital to France's nuclear industry - in northern Niger in 2010.
France currently has 4,000 troops in Mali, backed by 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.
