Fighting erupted in the divided Himalayan region after government forces raided a cluster of homes on a tip that the fighters were hiding in northwestern Sopore area, Muneer Ahmed Khan, police inspector general, said on Saturday.
He said that as the soldiers began searching homes, they came under gunfire from the fighters.
A police statement said the men belonged to the Lashkar-e-Taiba group. Police on Tuesday had killed one of the group’s senior Kashmir leaders, Abu Dajana.
As the news of the latest killings spread on Saturday, thousands protested and clashed with police in several parts of the region as residents chanted slogans against India and in favour of the fighters who have fought against New Delhi’s rule since 1989.
Soldiers fired at rock-throwing protesters in Bandipora area and wounded at least three civilians.
Tension has grown over the past week, with at least 10 fighters and four civilians killed in gun battles and in protests.
Two Indian army soldiers were also killed in an ambush by fighters.
India and Pakistan each administer part of Kashmir, but both claim the Himalayan territory in its entirety.
Armed groups demand that Kashmir be united either under Pakistani rule or as an independent country.
Anti-India sentiment runs deep in Kashmir’s predominantly Muslim population and most people support the fighters’ cause against Indian rule.
Nearly 70,000 people have been killed in the uprising and the ensuing Indian military crackdown.
In recent years, Kashmiris, mainly young people, have displayed open solidarity with anti-Indian fighters and sought to protect them by engaging troops in street clashes during military operations.
The anti-India protests and clashes have persisted despite the Indian army chief warning recently that “tough action” would be taken against stone throwers during counter attacks.
India accuses Pakistan of arming and training the fighters, which Pakistan denies.
Armed groups have largely been suppressed by Indian forces in recent years and public opposition to Indian rule is now principally expressed through street protests.