Airport officials told the BBC that flight operations are on hold and no employees, passengers or vehicles are being allowed to go towards the airport.
The attackers are still on the camp’s premises, holed up in a building, BSF officials said.
Police say that a body, suspected to be that of one of the attackers, has been found so far, the Press Trust of India reported.
The attackers’ identities have not been confirmed by officials. Indian news outlets say that the Islamist group, Jaish-e-Mohammad, has claimed responsibility.
Muslim-majority Kashmir is claimed in its entirety by both India and Pakistan, but the two countries control different parts of it.