Georges-Francois Leclerc, the prefect of the Haute-Savoie region, says initially 110 people were trapped, but 65 were airlifted out by helicopters.
The operation was suspended as night fell and will resume in the morning.
The cable cars are at "an altitude of about 3,800m (12,468ft)", officials said.
The problem was caused by cables that got crossed for "unknown reasons".
"We succeeded in evacuating, in very difficult conditions, 60 people in an hour-and-a-half," Mr Leclerc, told AFP news agency.
"We had to suspend the air transport rescue at nightfall," he added.
Local officials said the operation would resume at 06:30 local time (04:30 GMT) on Friday.
Mathieu Dechavanne, the head of the cable company, said earlier that the authorities were "in contact with the clients [in the cable cars]".
"They have water and some means of communication but are beginning to find it's taking a long time."
Italian and Swiss crews are taking part in rescue efforts. The operation involves three helicopters.
The cable cars offer tourists a panoramic view of Mont Blanc.
Forty-five people remain stuck on a series of cable cars high above the glaciers of Mont Blanc in the French Alps, a local official has said.