Adam Djaziri, a suspected Islamist who had been on the watchlist since 2015, was killed on Monday as his car smashed into a police vehicle on the French capital's most famous avenue.
Two handguns and a Kalashnikov-style assault rifle were found in the 31-year car, while a weapons stash was discovered at the home of the attacker.
A source close to the inquiry said a letter had been found in which Djaziri claimed allegiance to Islamic State group leader Abu Bakr al-Baghdadi.
In the letter, Djaziri boasted that he was playing a "double game" by amassing weapons for an attack by posing as a shooting enthusiast, the inquiry source said.
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The head of the French Shooting Federation said police officers had visited Djaziri's shooting club to enquire about him -- implying that his keen interest in guns had raised suspicions.
The attempted attack comes as France remains under a state of emergency after a wave of jihadist assaults that have left more than 230 people dead since 2015.
"What I know at this stage is that the first weapons permit was given before this individual was flagged up," he said in an interview with BFM television and RMC radio, but he added that "no one can be satisfied -- and certainly not me" that Djaziri was able to possess dangerous weapons after being put on a watchlist.
Interior Minister Gerard Collomb in an interview with Le Figaro newspaper to appear today explained that under current law the authorities cannot systematically take away a weapons permit from someone who is on a watchlist, but he vowed to take another look.
French Shooting Federation chief Philippe Crochard said Djaziri had been licensed for six years, and a source close to the probe said the attacker had requested a renewal of his permit in February.