Police arrested four men and one woman on Tuesday morning in Roanne in central France, and also in Rouen and Villeneuve-d'Ascq in the north (marked on the map below as 2, 1 and 3 respectively).
Officers seized two pistols and two machine guns, reported AFP.
The weapons belonged to a man close to one of the suspects, who held the guns legally.
Some of the suspects were in touch on social media or the internet and some of their messages made authorities fear they were planning violent acts, the source added.
The three raids were ordered by Paris prosecutors as part of a terror investigation, reported Le Parisien newspaper.
The arrests come after anti-terror police made at least ten arrests on Wednesday last week in both France and Belgium.
Various media report that the arrests were connected to the January 2015 terror attacks in Paris, where Amedy Coulibaly killed five people before he was shot dead by police.
France has been under a state of emergency since 130 people were killed in the November attacks in Paris in 2015.
There have been several other high profile terror attacks in France since then, most recently when a police officer was shot dead and two others were injured when a gunman opened fire on the Champs-Elysées in Paris two weeks ago.