Interior Minister Bernard Cazeneuve announced the sweep on Europe1 radio but did not specify if the arrests in the Paris region and southern France were linked to the suspect in the Brussels attack, Mehdi Nemmouche.
The minister said he would not elaborate further at this stage but a police source said three of detainees were suspected of recruiting fighters.
Nemmouche, 29, who was arrested on Friday in the southern French city of Marseille, is believed to have recorded a claim of responsibility for the May 24 Brussels attack in a 40-second video found in his possession, along with a Kalashnikov and a handgun.
Paris prosecutor Francois Molins said the "repeat offender" explains in the film that he had attached a GoPro camera to his bag to record his shooting rampage, but it had not worked.
Instead Nemmouche later "filmed his weapons and said he carried out the attack against the Jews in Brussels", said Belgian prosecutor Frederic Van Leeuw.
Nemmouche has so far said little during his detention.
Described as a "lone wolf" by the Paris prosecutor, Nemmouche became radicalised in prison and left for Syria on December 31, 2012, just three weeks after his release from jail.
He is believed to have fought there alongside fighters from the Islamic State of Iraq and the Levant (ISIL), and returned to Europe in March this year.
Some 780 people are thought to have left France to fight with jihadists in Syria, according to government estimates.
Cazeneuve on Monday evoked the problem of inmates being indoctrinated to become radical Islamists in French prisons, saying: "We have to react to this."
"As a start we can send properly trained imams who know the true spirit of Islam and its culture and who can go to prisons and explain this in prisons," he said.
A lone gunman entered the Jewish museum in the heart of Brussels last Saturday, removed an automatic rifle from a bag and opened fire through a door before leaving.
An Israeli couple and a Frenchwoman died on the scene and a 24-year-old Belgian man is in a critical condition.