The flight landed in Kandahar rather than Kabul because of high winds in the capital, Afghan sources said.
As reports of the bomb threat emerged, police and emergency vehicles had rushed to the airport in Urumqi, Xinjiang's provincial capital.
Last month Urumqi was the scene of the worst ethnic violence seen in China in decades when clashes between ethnic Uighurs and Han Chinese left at least 197 people dead and more than 1,700 injured.
KamAir was founded in 2003 as Afghanistan's first private commercial airline.
It serves a number of international and domestic routes with six aircraft and 350 employees, it says on its website.