Videos of the suspects show the alleged leader of a militant cell rapping about the September 11 attacks and another suspect saying he converted to Islam after a failed attempt to become a professional footballer in Britain.
Speaking after meeting top security officials, Prime Minister Jean-Marc Ayrault said the government was resolute in combatting extremism.
"I want to affirm the government's determination to fight against terrorism, as much inside our country as outside our borders," Ayrault said after meeting the interior and justice ministers to discuss the arrests.
"I know we can count on the support of the French people. We will not tolerate any form of racism or anti-Semitism," he said.
French authorities are on alert after a sweep at the weekend that left one man dead and 12 others in detention on suspicion of being involved in the bombing of a Jewish grocery or of planning other anti-Semitic attacks.
Interior Minister Manuel Valls has predicted more arrests and warned of possibly hundreds of home-grown militants at large.
The suspected leader of those detained, 33-year-old Jeremie Louis-Sidney, was shot dead Saturday after he opened fire on officers seeking to arrest him in a dawn raid at his home in Strasbourg.
France 2 public television has since broadcast a video of Louis-Sidney, reportedly recorded in May 2009, showing him rapping about his beliefs.
Dressed in a hooded winter jacket and wearing a Palestinian keffiyeh scarf, Louis-Sidney claims that "September 11 was just the tip of the iceberg".
"Know that you are being manipulated. If you don't understand, educate yourself," he raps in the video, ending with the phrase "Allahu Akbar" (God is Greatest).
In another video on YouTube, one of the suspects detained in the French Riviera city of Cannes, 18-year-old Yann Nsaku, says he converted to Islam after returning to France when an injury shattered his dream of playing football.
In the video, released on the website of newspaper Nice-Matin and reportedly made last December, Nsaku says football was his "passion" and that he played with the youth team of British club Portsmouth.
"I returned to France earlier than expected," he says in the video. "In France, I discovered Islam. Why did I choose Islam?... Simply because Islam answered all the questions I was asking myself."
Nsaku was among three suspects being held in Nice who were transferred to Paris Tuesday to be questioned by France's DCRI domestic intelligence service, police said.
President Francois Hollande vowed to step up security measures for the Jewish community after the arrests, which Jewish groups say came amid a worrying rise in anti-Semitic incidents.
France's SPCJ Jewish security watchdog said Monday that anti-Semitic acts surged by 45 percent in the first eight months of this year and were given new impetus by deadly attacks in March waged by Islamic extremist Mohamed Merah.
Merah went on a shooting rampage in and around the southern city of Toulouse, killing a rabbi, three Jewish children and three paratroopers before being shot dead in a police siege.
The new measures include a stepped-up police presence outside synagogues and other Jewish sites, but a police union representative raised concerns that officers lacked the resources to properly ensure security.
Frederic Foncel, the head of the SNPM union, said municipal police had been called in to protect sites in some cities instead of better-armed and better-equipped national police.
He said authorities were "lying" to Jewish groups about security by deploying municipal police who are "under-armed and not trained" to defend sites.
Foncel said the national police, who are equipped with bullet-proof vests and automatic weapons, would be in a better position to provide security.