The alert has been issued in the run-up to the Republic Day and US President Barack Obama's visit to India. Mr Obama will be the chief guest at the Republic Day celebrations.
Security has been tightened at the airports across the country after the alert.
An anonymous telephone call threatening that an Air India plane would be hijacked was made to the Kolkata office of the state-owned carrier on Saturday evening, police said.
Senior officials of Bureau of Civil Aviation Security, CISF and AAI held a meeting in Kolkata on Saturday night to review the security measures.
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