Ten people were arrested over the weekend across France, including in the region of Paris and the island of Corsica.
Their potential targets included radical imams, Islamist prison inmates who had been released or veiled women chosen at random, French daily newspaper Le Parisien reported.
Several firearms were discovered during the raids and some of the suspects had made grenades and explosives, the paper said.
The suspects are reportedly aged between 32 and 69, and include a retired policeman who is suspected of leading the ultra-right group.
A source close to the investigation said the suspects had an “ill-defined plan to commit a violent act targeting people of the Muslim faith,” the Agence France Presse news agency reported.
Another said they aimed to attack “targets linked to radical Islam.”
French interior minister, Gerard Collomb, thanked the country’s intelligence agency with a tweet, saying it watched "over every day the protection of the French against any violent action, wherever it comes from.”
Nikita Malik, director of the centre on radicalisation and terrorism at the Henry Jackson Society, said the alleged plots appeared to have some aspects of vigilantism, but cautioned against giving the groups any credibility.
She told The Independent the reported targets appeared to have been selected because "they are visibly Muslim in the way they dress".
This made them "easy targets," she said, adding: "It seems to be quite random."
Far-right groups are often driven by conspiracy theories "muddled with random facts they put in to put credibility to a wider narrative of white genocide and neo-Nazism," she said.
The methods used in far-right attacks often parallel those conducted by Islamist groups, for example the use of vehicles, knives and acid, she added.
"The themes and grievances that extremist groups use are very common. Even though they oppose each other, the methods they use are very common," she said. "They use low-tech for high impact. Particularly when they're operating in gangs, the resources at their disposal are often household resources or easy to obtain, in order to get the most leverage and coverage on the attack.
"Targeting veiled women is a very common thing that happens when we look at anti-Muslim hatred following an Islamist terror attack."
There was a similar wave of arrests in October 2017, in which the suspected targets were believed to be refugees and mosques.
Last year, a far-right terrorist targeted Muslims in a van attack outside the Finsbury Park Mosque in London, leaving one dead and 12 people injured.
He was jailed for life.