The Trinamool Congress chief said the car carrying three people, including two with a camera, sneaked into her convoy.
"It was then very late and there was no street light. This white car managed to get right behind my car in the convoy. It tried to hit my vehicle. We tried to avoid it and took a detour. Again the car came back.
"This happened four or five times. The policemen in our convoy asked them to stop, but the car did not. I asked the policemen not to fire," Banerjee said.
However, state police inspector general (law and order) Surajit Kar Purakayastha said that though they received a complaint, nobody was arrested till now.
Union minister of state for shipping Mukul Roy and state party president Subrata Bakshi, said that those were contract killers, who were sent to kill her.
"When the car was finally stopped, it was found that the driver was not carrying any driving licence. The two people carrying the camera also could not produce any press identity card," said an angry Trinamool chief.
However, Mamata herself seemed confused at the attempt.
"Why should somebody chase my car at 1am? And if the police now say that it was an intelligence bureau car, I will ask them what is the need for it? Already there are police escorts in our convoy. Are we insurgents or criminals?" she asked.
Roy accused the West Bengal home department and the state Communist Party of India (Marxist) headquarters of masterminding the attack.
"Already Buddhadeb and his government are trying to influence the probe," said Roy.