The Trinamool Congress chief said the car with three people, including two with a camera inside, sneaked into her convoy and tried to get very close to her vehicle when she started for home after a get-together.
"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.
When contacted, state Police Inspector General (law and order) Surajit Kar Purakayastha said: "We have received a complaint about this. It prima facie appears that it was a press car. Nobody has been arrested."
Banerjee, who was accompanied by Union Minister of State for Shipping Mukul Roy and state party president Subrata Bakshi, alleged that those inside the car could be contract killers sent to eliminate 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.
"Why should somebody chase my car at 1 am.? 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.
Later, Trinamool Congress trade union chief Purnendu Bose, who was travelling in another car in the convoy, filed a complaint at the Bidhannagar East police station.
Roy accused the West Bengal home department and the state Communist Party of India (Marxist) headquarters of masterminding an attack to kill Banerjee.
Chief Minister Buddhadeb Bhattacharjee holds the home portfolio. "Already Buddhadeb and his government are trying to influence the probe," Roy alleged.