Local media say the 39-year-old lived in the city's suburbs, and had been seen as a potential Islamist radical.
The gunman wounded two police officers before being shot dead by security forces on the Champs Elysees.
A pump-action shotgun and knives were found in his car, the French TV station BFMTV reported.
Another man suspected of possible links to the attack has turned himself in to Belgian police.
French police have also taken three family members of the gunman into custody, Reuters reports, citing a legal source.
Prime Minister Bernard Cazeneuve said security forces, including elite units, were fully mobilised ahead of Sunday's presidential poll.
"Nothing must be allowed to impede the fundamental democratic process of our country," he said after an emergency security cabinet meeting.
So-called Islamic State (IS) has said one of its "fighters" carried out the attack.