Prosecutors said 35-year-old Yassin Salhi, caused an explosion by ramming his car into an area containing flammable liquids.
His boss, the owner of a delivery firm, was found beheaded alongside flags with Arabic inscriptions.
Reuters reports that Salhi is being held in Lyon, where he was refusing to respond to interrogators on Saturday, according to a source.
His wife, sister and a fourth person are also in detention.
Salhi is known to have associated with Islamists for years and had previously been flagged by French authorities as a potential risk, but there has been no claim of responsibility for the attack.
While an anti-terrorist inquiry has been launched, Paris Public Prosecutor, Francois Molins, said it would be premature to make any conclusions at this stage.
”Questions remain over the exact chronology of events, what happened when he arrived, the circumstances of the decapitation, the motivation and whether there were accomplices,” he said.