The man, who attacked the police station on Thursday wearing a fake suicide vest, was said to be a Tunisian named Tarek Belgacem.
He was killed by officers as he ran towards the entrance of the police station shouting "Allahu Akbar" ("God is greatest"), exactly a year to the day since the massacre of journalists at the Charlie Hebdo newspaper.
Based on his fingerprints, police initially identified him as Sallah Ali, born in 1995 in Casablanca, a homeless man who was arrested for theft in 2013.
"This identity (he gave in 2013) is contradicted by a hand-written note that we found in his clothes," Molins told France Inter radio. "He is not known to the intelligence services under this name."
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Investigation sources told AFP that individuals claiming to be the parents and cousin of Belgacem have identified him from his photo.
"There is therefore a very strong indication that it is him, but it is still to early to speak of a formal identification," the source said.
His note was written in Arabic with a hand-drawn flag of the Islamic State group (IS).
The police station is in the 18th district of Paris, an area with a mainly North African population close to the tourist hotspot of Montmartre.
Describing the attack, an investigation source said the man pulled the cleaver from his inside coat pocket as he ran towards the officers. He "did not heed the warnings, and police opened fire".
A remote-controlled robot was also used to inspect the body for explosives.
A source close to the investigation said Thursday's attacker had pledged allegiance to IS leader Abu Bakr al-Baghdadi in the documents found on his body, and justified the attack as revenge for French bombings in Syria.