Abdullah Reda al-Hamahmy confirmed his name, his age of 29 and his Egyptian nationality to investigators after initially refusing to speak, the source said.
Investigators believe Louvre attacker is Egyptian
He was carrying spray paints in his backpack and said he wanted to damage the paintings at the world-famous museum, the source said, adding that the suspect's statements should be taken with caution.
The suspect also spoke of wanting to "avenge the Syrian people", the source said, referring to the civil war in that country in which hundreds of thousands of people have been killed and millions displaced.
Hamahmy was shot several times on Friday after attacking soldiers as he cried "Allahu Akbar" in what French President Francois Hollande described as a terrorist attack.
He has been in hospital since the attack and late on Tuesday his police custody was lifted as his health had "greatly deteriorated" during the day, the source said, without elaborating.
Knife-wielding man shouting 'Allahu Akbar' shot by soldier in Paris
France, which is preparing for a presidential election in April and May, remains under a state of emergency following a series of attacks by militant Islamists over the past two years in which more than 230 people have been killed.
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