The incident took place early yesterday on Paris's iconic Place de la Concorde, where the world's most famous bike race reaches its climax.
A car crashed through the security barriers several hours before the riders were due to come through and police opened fire on the vehicle as it sped away from a checkpoint.
Officials said they believed there was no link to the Tour de France itself.
"Investigations are underway to see if he was at the wheel," a legal source told AFP.
Another person has also come forward also claiming to have been in the car when the incident took place. Both have been remanded in custody.
And a young woman, believed to have been in the car, checked herself into a hospital in the western Paris suburbs with bullet wounds to the chest.
She was operated on overnight and "her life is not in danger," said a source close to the investigation.
"The probe will determine if the bullet that hit the young woman came from a police shot," said this source.
In triumphant scenes later yesterday, British cyclist Chris Froome won his second Tour de France on the rain-lashed Champs Elysees, with Germany's Andre Greipel clinching a thrilling final sprint.
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