Anti-terrorist officers from the DST domestic security service swooped on the suspects in dawn raids in the Venissieux suburb of Lyon on Tuesday.
Chellali Benchellali, the imam of Venissieux mosque, was detained along with his wife Hafsa and one of their sons, Hafed, as well as up to five others.
"(The arrests) will act as a brake on all these networks and this radical and political Islam which is doing so much harm in our suburbs," Venissieux mayor Andre Gerin, a communist, said.
Another of the imam's sons, Mourad Benchellali, was previously arrested by US forces in Afghanistan and is being held at the US military base at Guantanamo Bay, Cuba.
Around 10 people, mostly of Algerian origin, are already in detention as part of the Chechnya-related investigation launched by French magistrates in December 2002. Among the 10 is another of Chellali Benchellali's sons, Menad.
The 10 are accused of planning attacks on Russian interests in France, including the Russian embassy in Paris, to avenge the death of an Arab-born Chechen fighter killed by Russian troops.
"(The arrests) will act as a brake on all these networks and this radical and political Islam which is doing so much harm in our suburbs"
Fouad Alaoui, of the Union of Islamic Organisations of France, told Aljazeera.net that arrests of Muslims for political reasons are rare in France.
He added that it was too early to comment on this particular incident, and would have to wait for the facts to emerge.
But his organisaton has claimed in the past that a racist, Islamophobic climate exists in France, and Arabs and Muslims are often its victims.
Russia has been intermitently fighting a war against the Muslim Chechens since the Soviet Union collapsed in 1991.
Russian troops marched back into Chechnya in 1999 after a 1994-96 war ended in de facto independence for the region.
Support for the Chechen cause is strong amongst France's Muslims, as it is throughout Muslim communities all over the world.