The bus carrying more than 50 passengers was on a narrow mountain road when it veered off and plunged about 100 metres near Trungal in the Assar area.
Police and rescue workers rushed to the site in the Himalayan region.
Federal minister Jitendra Singh said helicopters were being arranged to take the injured to hospitals.
“Unfortunately, 36 persons have died and 19 injured, out of whom 6 injured are serious,” Mr Singh said on X, formerly Twitter.
“All possible help being provided … helicopter services to be arranged for the shifting the more injured ones.”
Prime Minister Narendra Modi announced compensation of 200,000 rupees ($2,400) to families of the victims and assured treatment for the injured.
Footage of the accident site shared on social media showed the mangled remains of the vehicle lying in the gorge as locals and rescue workers pulled the injured from the bus.
Jammu and Kashmir, close to the border with Pakistan, is notorious for road accidents due to its treacherous terrain and narrow roads as well as overcrowding of public transport buses.
At least 10 people died and 55 were injured after a bus skidded off a motorway into a gorge near Jammu in May. At least 12 passengers were killed after a bus fell into a gorge in region’s Poonch district in September last year.