An investigation is now taking place into a "criminal terrorist association", a judicial source said.
Details of the suspects and the alleged plot have not yet been released.
"This investigation is looking into a plot, vague and ill-defined at this stage, involving violent action against the president of the republic," prosecutors said on Tuesday.
The suspects had reportedly been under surveillance by France's domestic security agency, the Direction Generale de la Securite Interieure (DGSI).
The arrests were carried out in the eastern French regions of Moselle and Isere and the northern region of Ile-et-Vilaine, officials said.
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