Officer Abdul Wakil said the accident happened on Sunday in the Chilas district, on the route from Skardu to Rawalpindi, a city in the country’s northwest.
Reports suggest up to 26 people have been killed, and 15 others have been injured.
Mr Wakil said the remote mountainous terrain has made rescue efforts difficult, and that workers do not have the necessary equipment and resources.
Amiruddin Mughal, in Pakistan, told Sky News that 37 passengers were on the bus at the time – including women, children, and soldiers.
Such accidents are fairly common in Pakistan, where motorists largely disregard traffic rules and battered roads are poorly maintained.
Last month, a speeding bus fell off a mountainous road into a river, killed 24 passengers.