Investigators are still questioning the parents and a friend of the man who attacked passers-by with a knife in central Paris.
The 20-year-old Frenchman, born in the Russian republic of Chechnya, killed one person and wounded four others on Saturday before police fatally shot him. The Islamic State group claimed responsibility.
Khamzat Azimov's parents and a friend from the eastern city of Strasbourg were being detained by police.
A judicial official said on Monday the suspect was living in the northern 18th district of Paris with his family, which had previously lived in Strasbourg.
Counter-terrorism investigators want to know if the assailant had help or co-conspirators.
The attack has raised fresh questions about how France's radical watch list is used. French authorities had revealed that the suspect was on a police watch list for radicalism, but he had a clean criminal record.
Conservative leader Laurent Wauquiez criticised the government's "blindness" and "inaction" following the attack. His Republicans party called on centrist president Emmanuel Macron to take measures to "preventively intern the most dangerous individuals" listed as radicalised.
Far-right leader Marine Le Pen asked what the purpose of the list was if it was not used to stop potential attackers.
French authorities have previously explained that the register, designed as a tool for intelligence services, contains the names of thousands of individuals suspected of being radicalised but who have yet to perform acts of terrorism.
French interior minister Gerard Collomb will hold a meeting with France's prefects in charge of defence and security later.
Azimov obtained French nationality in 2010. He was born in the largely Muslim Russian republic of Chechnya, where extremism has long simmered. Chechens have been among the numerous foreign fighters in Syria and Iraq, some joining the Islamic State cause early in the fighting.
French media report the victim killed in the attack was a 29-year-old man identified only by his first name, Ronan, who was living in the 13th district of Paris. One of his neighbours told reporters he was a "very smiling" man, with a "great generosity".
