Local police and witnesses said a grenade thrown inside a tourist bus in Srinagar killed six people and wounded 15 others.
Another blast outside a tourist reception centre in Srinagar wounded six.
Shabir Ahmad, a shopkeeper who saw the attack on the bus, told Reuters: “I saw a huge orange flash coming out of the bus with a loud explosion.”
Television channels said some of the tourists killed and wounded were from the Indian state of West Bengal.
A second grenade attack in Srinagar’s busy Regal Chowk injured four people who were travelling in a car registered outside Kashmir, police said.
In two more attacks on private passenger jeeps near Lal Chowk, two tourists were killed and ten people were injured.
No group has claimed responsibility for any of the five attacks.
Decrease in violence
Many tourists visit the Himalayan region each year, despite fighting in which more than 45,000 people have died since 1989.
Violence between India and Pakistan has decreased since 2004, and more than 600,000 tourists visited Kashmir last year.
Muslim fighters have been attacking visitors since the start of the tourist season in April.
At least 21 people were wounded on May 31 in a grenade attack on a tourist bus near Dal Lake.
Muzaffar Baig, Kashmir’s deputy chief minister, told a TV channel: “We think there is a concerted effort which has some support from across the border to not only create a state of terror in the state but also destroy the industry of tourism.”
India has accused Pakistan of arming and training militants to wage a revolt in its part of Kashmir. Islamabad has denied this.