"The suspects had a plot that was sufficiently advanced for the police to decide to arrest them," a source said.
In Clermont-Ferrand a bomb squad was called in to sweep the home of a 37-year-old suspect, another source said.
The raids were ordered by anti-terrorism prosecutors in Paris, a judicial source told the AFP news agency.
France, which has been rocked by a wave of deadly jihadist attacks from January 2015, remains on high alert.
Louvre attack, Montpellier arrests
On 3 February a 29-year-old Egyptian armed with machetes lunged at four soldiers on patrol outside the Louvre museum, shouting "Allahu Akbar". He was shot and seriously injured by one of the soldiers.
A week later, police arrested four people in southern France, including a 16-year-old girl near Montpellier, on suspicion they were planning an "imminent" attack.
During the raid, the officers found ingredients used to make TATP, a highly unstable homemade explosive used in the November 2015 attacks in Paris.
The teenage girl, her partner and an older man, described as their mentor, were charged with terrorist offences.