The Socialists have declared they will withdraw their slates in Nord-Pas-de-Calais-Picardie in the north east of France and in Provence-Alpes-Côte-d'Azur in the south east.
The party leadership also called for the slate to be withdrawn in Alsace-Lorraine-Champagne-Ardenne but local chief Jean-Pierre Massenet has refused to comply, prompting a sharp rebuke from national secretary Jean-Christophe Cambadélis.
FN leader Marine Le Pen, who heads the party's slate in the North, described the Socialists' decision to withdraw as "collective suicide" and accused them of treating their voters like sheep to be taken to sacrifice.
"Those voters will have no representatives for six years," she pointed out.
The executive of former president Nicolas Sarkozy's Republicans, formerly the UMP, on Monday endorsed his announcement that it would not merge lists with the Socialists or withdraw its own.
Only two members of the party's political bureau, ex-ecology minister Nathalie Kosciusko-Morisot and former prime minister Jean-Pierre Raffarin, opposed Sarkozy's line, arguing that the Republicans should withdraw in the Languedoc-Roussillon-Midi-Pyrenées region, where they are in third place after the FN and the Socialists.
Sarkozy came under fire for his leadership of the party, however.
Hervé Mariton, one of his rivals to be the party's presidential candidate in 2017, said that Sunday's results showed he was "not credible" to voters, having lost the 2012 presidential poll.
Another aspiring presidential candidate, former prime minister Alain Juppé, supported non-withdrawal but, according to sources, made it clear that he might not do so in future elections.
And former labour minister Eric Woerth, who has been a close ally of the former president, declared that the mainstream right was not ready for this poll and did not have an "official and legitimate leader".
Unsurprisingly, the far right was on the offensive on Monday.
Marine Le Pen declared that the party would cut off all international aid paid by the North region if it wins next Sunday, saying it was not the regional council's role to buy "mosquito nets for prisoners in Senegal".
The region spends six million euros of its 1.7-billion-euro budget on projects in Senegal, Mali, Madagascar, Brazil and the Polish region of Silesia, according to officials.
He congratulated his daughter and his niece, Marion Maréchal Le Pen, who headed the slate in Provence-Côte d'Azur, saying he was certain they would win control of the two regions.
"With 40 per cent of the vote you can take power," he commented.