It has been reported that al Qaeda-linked rebel fighters fled the town as French and Malian forces swept north after retaking Gao and Timbuktu earlier this week.
Haminy Belco Maiga, president of the regional assembly of Kidal said: "The French arrived late at night and deployed in four planes and some helicopters."
He added he had seen no early indications of resistance.
French Armed Forces spokesman Thierry Burkhard confirmed in Paris that French troops were in Kidal and said they had taken control of the airport. "The operation is continuing," he said, declining to give further details.
It was not immediately clear whether the French troops were accompanied by Malian forces.
Tuareg MNLA rebels who want greater autonomy for the desert north said earlier this week they had taken control of Kidal after Islamists abandoned the town.
The MNLA, which fought alongside the Islamists before being sidelined by them in mid-2012, was not immediately available for comment on the French deployment.
Kidal is the capital of a desert region with the same name into which Islamist fighters are believed to have retreated during nearly three weeks of French air strikes and a joint advance by thousands of French and Malian ground troops.
The offensive in France's former West African colony is aimed at heading off the risk of Mali being used as a springboard for jihadist attacks in the wider region or Europe.
French troops now number 3500 on the ground,
Doubts remain about just how quickly an African intervention force, now expected to exceed 8000 personnel, can be fully deployed in Mali to track down retreating al Qaeda-allied insurgents in the north.
French Foreign Minister Laurent Fabius said the French operation was planned to be a lightning mission that would last just a few weeks to avoid getting bogged down.
"Liberating Gao and Timbuktu very quickly was part of the plan. Now it's up to the African countries to take over," he said.
"We decided to put in the means and the necessary number of soldiers to strike hard. We will leave very quickly."
But Mr Fabius warned things could now get more difficult.
"We have to be careful. We are entering a complicated phase where the risks of attacks or kidnappings are extremely high. French interests are threatened throughout the entire Sahel," he said.
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