According to BBC Turkish, Georgia’s aviation authority said the aircraft disappeared from radar without issuing a distress signal.
Georgia’s interior ministry said the plane crashed near Sighnaghi, around 5km (3 miles) from Azerbaijan.
Azerbaijan’s President Ilham Aliyev said in a message to Erdogan he was “deeply saddened by the tragic news of the loss of servicemen”, and extended condolences to him and the people of Turkey.
The US ambassador to Turkey, Tom Barrack, said on X he was “deeply saddened” by the crash.
“We extend our heartfelt condolences to the families of the fallen, the Turkish Armed Forces, and the people of Türkiye,” he added.
The C-130 is a US-made transport plane used by air forces around the world.