Most of the dead found after the avalanche on Mont Maudit, which translates as "Cursed Mountain", were Europeans, police said.
At least two Germans, two Swiss and two Spaniards, were initially found dead after the early morning avalanche, local police Colonel Bertrand Francois said, while another three bodies were discovered during rescue efforts.
Authorities had said earlier that five Britons and two Spaniards were among those missing but it was unclear if the bodies found were theirs.
Nine people were injured in the disaster.
Efforts were continuing to track down the missing climbers on the mountain in the Mont Blanc massif. Francois said they had been moving ahead of the rest of the group so may have avoided the avalanche.
In Paris, Interior Minister Manuel Valls said he was heading to the scene to oversee operations.
"Searches are still underway to find the missing," he said in a statement.
"The interior minister wants to assure the families of his deep sympathy and full support."
One of the injured sounded the alert at around 0325 GMT after the avalanche on Mont Maudit, the massif's third-highest peak, rising to an altitude of 4,465 metres (14,650 feet), and considered one of the more difficult paths to climbing Mont Blanc.
"No weather report was forecasting an avalanche risk," the mayor of the nearby town of Chamonix, Eric Fournier, told AFP.
It is the deadliest climbing disaster in at least a decade in France.
In August 2008, eight climbers -- four Germans, three Swiss and an Austrian guide -- were swept away after blocks of ice broke off Mont Blanc du Tacul, prompting an avalanche.
Thousands of tourists flock to the French Alps every year for sports including mountain climbing and skiing, but every year some fall victim to accidents.
A Norwegian cross-country skier died in April after being caught up in an avalanche on Mont Blanc, only about a month after a Canadian skier died after plunging into a 20-metre (65-foot) crevice on the mountain.
Rescue workers are often called in to assist stranded climbers or skiers.