The cable connects the Aiguille du Midi station in France to the Italian Punta Helbronner.
It is thought the trapped tourists were suspended 3700m in the air.
The route attracts around half a million tourists each year.
Italian news service R.it reports that strong winds may have caused the cable cars to stop.
The winds may have made the wires which support the cabin cars cross over.
Eric Fornier, the Mayor of Chamonix, told BFMTV that rescuers, supported by their Italian counterparts, are working to bring the trapped people to safety.
And Frederic Maurer, 49, one of the first passengers to be rescued along with his daughter and son, said: "We were in the cab for two-and-a-half hours locked under the sun.
"We had just started from flagship station Helbronner on the Italian side, when everything froze."
Mathieu Dechavanne, chief executive of the company that manages the cable cars, said the weather in the area was good, but the rescue operation would "take a while".
He told La Vanguardia: "We are in contact with customers. They have water and means to communicate."