The Russian news agency Interfax reports, citing Kazakhstan's Vice Prime Minister Kanat Bozumbaev.
The plane did not crash straight down, but caught fire when the pilot lost control during the emergency landing.
Images from the scene show how large parts of the cabin were virtually destroyed by the fire, but the tail section fared much better. The rescue teams initially found six survivors, but the number gradually increased to around 30, according to regional healthcare authorities. Some are described as severely injured and are receiving intensive care at the hospital.
The plane from Azerbaijan Airlines was supposed to fly from Baku in Azerbaijan to Grozny in Russia. However, for some reason, it was redirected, first to Makhachkala in Russia, then to Aktau in Kazakhstan, on the other side of the Caspian Sea.
After crossing the sea, the plane attempted to make an emergency landing three kilometers from the coastal city of Aktau.
According to Azerbaijan's President Ilham Aliyev, the plane changed its planned route due to deteriorating weather conditions.
On board the plane were passengers from Azerbaijan, Russia, Kazakhstan, and Kyrgyzstan, according to Kazakh authorities.