Investigators say tire marks at the accident site suggest that the bus was traveling too fast and the driver may have made a mistake in steering the wheel.
Kangra SP Santosh Patial said that among those killed also included the 67-year-old driver Madan Lal and two women teachers.
The bus was transporting children, all below the age of 13, who were studying at a private school in Nurpur town, 60km (37 miles) south of Dharamsala.
The young victims were said to be aged between 10 and 12. Two of them were in a critical condition.
Another 15 people were taken to nearby hospitals for treatment. The bus plunged into a deep gorge in Himachal Pradesh, India. There were 40 to 45 people on the bus, including school staff and students. The bodies were rescued with the help of local people.
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India has the highest number of road fatalities in the world. Fatal accidents are common.
Some reports have indicated the bus driver lost control of the vehicle and was driving recklessly.
Prime Minister Narendra Modi's office tweeted that Mr Modi was "anguished" over the news of the deaths.
"At this time, Omaha Public Schools administrators, Student Transportation of America and OPS Student Transportation are cooperating with the Omaha Police Department as they investigate the incident", according to a statement from OPS early Monday evening.
The government has said the families of the children killed will receive 5 lakh Indian rupees ($7,700 USD). "My prayers and solidarity with those who lost their near and dear ones in the accident", he said in a tweet.