He was "ashamed not to have killed" 35-year-old teacher Benjamin Amsellem, according to a police source quoted by the paper, who said that he became interested in radical Islam after watching a television reports that "more or less put forward the hypothesis that Westerners were being persecuted by the Muslims".
He reportedly declared his allegiance to the Islamic State armed group, while explaining that he does not "represent" them.
Two MPs, Claude Goasguen of the right-wing Republicans and Meyer Habib of the centre-right UDI, wore kippas briefly when they arrived at the National Assembly on Wednesday out of solidarity with the victims of anti-Semitism.
Habib, who represents French nationals living abroad, has joint French and Israeli citizenship.
Their gesture followed a call by Marseille's top Jewish leader, Zvi Ammar, for Jewish men and boys to stop wearing the skullcap "until better days".
"Unfortunately for us, we are targeted," he said on Tuesday. "As soon as we are identified as Jewish we can be assaulted and even risk death."
Amsellem was wearing one at the time of the attack and there have been four other attacks on Jews in the city in the last three months.
There were 84 per cent more reported anti-Semitic attacks in France in January-May 2015 than in the same period the previous year and they included the murder of four people in a kosher supermarket in the wake of the January's attack on Charlie Hebdo magazine.
France's Chief rabbi Haim Korsia on Tuesday acknowledged that Ammar was motivated by an "understandable emotion" but insisted, "We should not give an inch."
Government spokesperson Stéphane Le Foll declared that he state must ""provide protection and condemn anti-Semitic acts" but said that the debate on wearing the kippa "originated within the Jewish community".
Education Minister Najat Vallaud-Belkacem was more forthright, saying: "It's certainly not the advice I would give, personally."
Right-wing MP and former minister Brice Hortefeux agreed with the chief rabbi that "giving up [the kippa] is giving in" but said it was impossible "not to modify your behaviour in the face of these unspeakable acts".