Four of the suspects were rounded up by anti-terror police in the eastern Moselle département, another in Ile et Vilaine in western France and the last in Isère in the French Alps, sources told various French media.
The plan is "imprecise and ill-defined at the moment", the sources said, and the investigation aims to establish how real the threat was and its exact nature.
On Monday, at a ceremony to welcome the new head of the DGSI anti-terror unit, Interior Minister Christophe Castaner said he was "closely following" threats from "extremist movements of both right and left, which are very active on our soil".
Previous plotters arrested
In October 2017 police broke up a group organised by Logan Nisin, a resident of the southern city of Vitrolles, which aimed to "trigger remigration through terror".
It had listed Castaner and hard-left leader Jean-Luc Mélenchon as possible targets for attack.
In June, 13 members of another group, including their probable leader, a retired police officer, were charged with organising a "terrorist gang".
They planned to "fight the Islamist peril" by attacking Islamists as they left prison or radical mosques.
Four of them, including the leader, were freed under judicial control, against the advice of the public prosecutor.
The small groups the DGSI classifies as "ultraright" have limited operational capacity, according to a report filed during the Logan Nisin case.