The agencies Ifop, Opinion -Way and Ipsos base their projections on actual vote count in select constituencies.
They projected that the National Front won between 27% and 30% of support in the voting, followed by former president Nicolas Sarkozy’s Republicans party and president François Hollande’s governing Socialists.
The elections took place in an unusually tense security climate, expected to favour conservative and far right candidates and strike a new blow against the Socialists.
While National Front had significant support, the party may not be able to translate that into victory in the second-round voting on December 13 for leadership of France’s 13 newly-drawn regions.
Mr Sarkozy’s Republicans party and its allies were projected to come in second place at around 27%.
The Socialists, who run nearly all of the country’s regions, are projected to come in a weak third place, with between 22 and 24%.
The Paris attacks on November 13 that killed 130 people, and a Europe-wide migrant crisis this year, have shaken up France’s political landscape.
The National Front’s leader, Marine Le Pen, is hoping the two-round voting that started yesterday will consolidate political gains she has made in recent years — and strengthen the party’s legitimacy as she prepares to seek the presidency in 2017.
The unpopular Mr Hollande has seen his approval ratings jump since the Paris attacks, as he intensified French air strikes on Islamic State targets in Syria and Iraq and ordered a state of emergency at home. But his party, which currently runs nearly all of France’s regions, has seen its electoral support shrivel as the government has failed to shrink 10% joblessness or invigorate the economy.
Many political leaders are urging apathetic voters to cast ballots as a riposte to fundamentalists targeting democracies from France to the US.
First-time voter Eli Hodara, 18, expressed hope that more young people would turn out. “I think it is important to vote, even if one leaves the ballot blank,” said the Paris student.
It is the last election before France votes for a president in 2017, and a gauge of the country’s political direction.
“It’s an important moment; important for the future of our regions, important also for the future of our country, important with regard to the catastrophic and dramatic events that have hit France,” Ms Le Pen said as she cast her ballot in Henin-Beaumont.
The National Front denounces Europe’s open borders, what it calls the “migratory submersion” and what it claims is the corrupting influence of Islam on French civilisation.