"France has attacked Islam. We will strike at the heart of France," said Abou Dardar, a leader of Movement for Oneness and Jihad in West Africa (MUJAO), an offshoot of Al Qaeda in the Islamic Maghreb (AQIM), speaking to AFP by telephone.
Asked where they would strike, he said: "Everywhere. In Bamako, in Africa and in Europe."
The French offensive has blocked the advance of Islamist forces towards the capital Bamako from their bases in the north which they have controlled since last April.
On Sunday, French aviation struck at targets in the central Islamist strongholds of Gao and Kidal.
Sixty Islamists were killed in Gao alone on Sunday, according to residents and a regional security force.
The MUJAO official also referred to France's eight hostages held in the Sahel region.
"We will make a statement on the hostages today. From today all the mujahedeen are together."