The suspects, aged 17-25, were rounded up on Tuesday in south-east France and the Paris region.
Nine of them are male, of whom three are minors, according to sources.
The 10th is reported to be the mother of Logan Alexandre Nisin, who was arrested in the southern town of Vitrolles for terrorist conspiracy on 28 June.
His 10 associates face the same charge after an investigation uncovered "intentions to commit violent acts", whose details remain unclear, a judicial source told the AFP news agency.
Admirer of neo-Nazi mass murderer
Nisin, who had no steady job and lived with his mother, came to the police's attention because of a Facebook page he administered that glorified neo-Nazi Anders Behring Breivik, who killed 77 people at a Norwegian Labour Party youth camp in 2011.
Investigators have established that he possessed arms, including a pump-action shotgun, and intended to act on his threats to kill immigrants, drug-dealers and jihadists, sources say.
He had established an organisation he called OAS - the initials of the secret armed organisation that resisted French withdrawal from Algeria in 1961 - whose declared aim was to "set off remigration".
It named potential targets including an Indian restaurant in Aix-en-Provence, a flea-market in Marseille and the two politicians, according to Le Monde newspaper, which published an inquiry into Nisin on the day of the latest arrests.
Neither Castaner, who was until recently mayor of a town in Provence that Nisin regarded as lax on immigration, nor Mélenchon had been told that they might be targets, they both revealed on Wednesday. Mélenchon's supporters said that a request for protection during the parliamentary election campaign had been turned down.
Since the age of 16, Nisin has joined a series of far-right groups, notably Action Française, a monarchist organisation founded in the 1930s by Charles Maurras, an anti-Semitic writer who was jailed after World War II for collaborating with the German occupation.
The organisation has gained a new lease of life in the Marseille area recently, its young members attacking students campaigning against the previous government's labour reform and other left-wingers.
After his arrest in June, Action Française issued a statement pointing out that he was no longer a member.
With the debate on a new anti-terror law, set to be passed by the French parliament on Wednesday, centring on Islamist violence, the latest plot serves as a reminder of a warning by former intelligence chief Patrick Calvar, who retired from the post in June.
"We should sooner or later devote more resources to look into other extremist groups because confrontation between the ultraright and the Muslim world is inevitable," he told a parliamentary committee in 2016.