The incident happened on the ancient Pont Neuf, in the centre of the city, soon after Emmanuel Macron’s re-election as president on Sunday night.
“A car came rushing towards police soon after midnight, and refused to stop,” said an investigating source.
“One of the police officers opened fire, killing two people and wounding a third.”
In the early hours of Monday, an examining magistrate attended the scene, which was closed to traffic.
The IGNP investigations unit – one that deals with potential crimes carried out by the police – also attended.
Police at the scene confirmed the wounded person was a woman who had been travelling in the back of the car.
She was rushed to the Pitié-Saplêtrière hospital “for urgent treatment,” said the investigating source.
All of the shots were thought to have been fired by a police-issue Heckler & Koch G36 assault rifle.
There was no return of fire from those inside the car, none of whom was thought to be armed.
“It seems that they were simply in a car that was being driven erratically,” said the source.
There were no early clues pointing to a link with the presidential election, or any terrorist acts.