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The E.U. has become a bureaucratic machine that excels at dispensing edicts on how cheese is labeled while ignoring everyday problems such as unemployment and illegal immigration. By some polling estimates, myriad anti-EU groups could nearly double their tally in May 22-25 voting from the last European Parliament election five years ago by taking as much as a quarter of the seats.

Casting the EU as public enemy No. 1, Ms. Le Pen and other nationalists are presenting themselves as credible alternatives to Europe's mainstream, pro-EU leaders, and no longer as mere loudspeakers for protest voters. "The EU nowadays is like the U.S.S.R.: It can't be improved. We need to let it crumble and build after it a Europe of free and sovereign nations," Ms. Le Pen said in an interview at the National Front headquarters just outside Paris.

Having expanded her National Front's following in France, she is setting out to unite Europe's disparate nationalist parties into an anti-EU caucus at the European Parliament—one that could stall the decadeslong march toward a United States of Europe.

The formation of a potent anti-EU minority would also pose a risk to EU policies some economists consider important to restoring growth. One likely target: an ongoing effort to forge a trans-Atlantic free-trade agreement. That wouldn't bother Ms. Le Pen. She calls the effort "pure folly."  As the election approached, mainstream politicians were sounding the alarm. "The ideas promoted by the far right aren't the values of France and once were behind Europe's nightmare," said French Prime Minister Manuel Valls, alluding to the world wars. In Italy, President Giorgio Napolitano warned of "populist impulses" that endanger the EU. With many EU members' economies still limping, the bloc has yet to demonstrate an ability to halt the ravages of the sovereign-debt crisis, making this election a crucial test.