The five men and one woman, aged 22 to 62, were seized by France's DGSI domestic intelligence agency, which did not release their identities, a source close to the case has told AFP.
Four suspects were picked up in the northeastern city of Moselle, a little more than an hour's drive from Verdun, which Macron was visiting Tuesday as part of weeklong commemorations of the World War I centenary.
Another was detained in southeast France and a sixth in the west.
An inquiry opened by the Paris prosecutor's office, in charge of terrorism investigations across the country, is focusing on "a vague and ill defined project at this stage," one of the sources said.
On Monday, Interior Minister Christophe Castaner had warned that France remained "on alert" over the threats posed by "extremist networks both on the right and left, which are quite active in our country".
It was not the first known attempt against Mr Macron.
In July last year, a 23-year-old was charged with plotting to assassinate the president at France's Bastille Day military parade.
The man told investigators he wanted to kill Mr Macron along with "Muslims, Jews, blacks and homosexuals," and three kitchen knives were found in his car.
Several other suspected far-right members have been arrested since 2017 - the year Mr Macron was elected president - after police found they might be preparing imminent attacks.
During the French summer 13 people with links to the radical Action des Forces Operationnelles (Operational Forces Action) group were arrested by anti-terrorist police over an alleged plot to attack Muslims.
And 10 people were also arrested in October 2017 over suspicions of wanting to target Muslims as well as politicians like the far-left firebrand Jean-Luc Melenchon as well as Castaner.
In a report filed by the DGSI as part of that investigation, it said the far right in France consisted of several disparate and competing groups with only "limited" operational capacity.
