The vehicle was driving against traffic and sped towards to the officers, who were forced to open fire, according to police sources.
The incident happened on the Pont-Neuf in the heart of the city, and early reports say that police attempted to stop a vehicle that was driving the wrong way.
A large police presence descended upon the scene in the heart of the capital, where white sheets covered the bodies and a Volkswagen sedan could be seen bisecting traffic lanes, according to AFP journalists and photographs.
Less than two kilometres away, French President Emmanuel Macron was celebrating his victory over far-right rival Marine Le Pen with a crowd of supporters on the Champ de Mars park.
No link has been established between the bridge incident and the election night events.
An Egyptian tourist, who gave his name as El Sammak, said he was near the incident on the terrace of the White Horse Hotel overlooking the Seine when he heard the shooting.
“I heard four bullets (being fired),” he told AFP. “When I looked, I saw a man running 10 or 15 metres. Then he collapsed. Apparently, he was not the driver but a passenger.”
An investigation has been launched for “attempted voluntary manslaughter on persons in charge of public authority”.
The Paris public prosecutor, Laure Beccuau, arrived around 1.30am.
France’s police oversight body, the General Inspectorate of the National Police, is also expected to be deployed, as is typical following an officer’s use of a firearm.