The accident occurred near Khaneri in Rampur, some 140 km from the state capital Shimla, a famous tourist spot.
The bus with 35 passengers on board was on its way from Rekong Peo town in Kinnaur district to Nauni in Solan district when it skidded off the road and crashed into the gorge.
"Most of the dead belonged to Rampur town and its nearby areas," a senior police officer said. The injured were rushed to a hospital.
The authorities, who are carrying out the rescue operations, have not ruled out the possibility of higher casualties as the bus rolled down around 200 metres towards the bank of the Sutlej River.
Condoling the loss of lives in the bus accident, PM Narendra Modi tweeted:
Last week, 16 pilgrims died in a bus crash in neighbouring Jammu and Kashmir state, while in April 44 people were killed in Himachal Pradesh in a similar accident.
My thoughts are with those who lost their loved ones in the tragic bus accident in Himachal Pradesh. May those injured recover quickly: PMPMO India (@PMOIndia) July 20, 2017