Bachchan had come to the city at the invitation of the state government for the inaugural programme of the 23rd Kolkata International Film Festival, which opened on November 10.
The superstar was on his way back to the airport on November 11 when the incident occurred.
He was accompanied by a senior state minister, the official told PTI yesterday.
The car had been provided by a travel agency.
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The government has sought an explanation from the agency, the official said, adding that it had been paid a "huge amount" for transporting the actor.
The official said an initial probe has revealed that the vehicle's certificate of fitness had expired but the car was still in use.
A source in the secretariat said "proper action" may be taken against the transport agency if it is found to be at fault.