The RN was seen winning about 34 per cent of the vote, exit polls from Ipsos, Ifop, OpinionWay and Elabe suggested, ahead of President Emmanuel Macron's Together alliance which appeared to have 20.5-23 per cent and the progressive New Popular Front projected to win about 29 per cent.
The two-round elections that wrap up on July 7 could affect European financial markets, regional support for Ukraine and the management of France's nuclear arsenal and global military force.
Many French voters are frustrated about inflation and other economic concerns, as well as Macron's leadership, seen as arrogant and out-of-touch with their lives.
The National Rally (RN) party has tapped into that discontent, notably via online platforms like TikTok, and led in pre-election opinion polls.
A new progressive coalition, the New Popular Front, also poses a challenge to the pro-business Macron and his centrist alliance.
It includes the French Socialists and Communists, the greens and the hardline France Unbowed party and vows to reverse an unpopular pension reform law that raised the retirement age to 64, among other economic reforms.
The National Rally hopes to make gains that will lead to Jordan Bardella becoming prime minister. (EPA PHOTO)
There are 49.5 million registered voters who will choose the 577 members of the National Assembly, France's influential lower house of parliament.
Turnout stood at an unusually high 59 per cent with three hours to go before polls close - 20 percentage points higher than turnout at the same time in the last first-round vote in 2022.
Some pollsters suggested the high turnout could temper the outcome for the RN, possibly indicating voters made an extra effort to cast ballots for fear that it could win.
The vote was taking place during the traditional first week of summer holidays in France, and absentee ballot requests were at least five times higher than in 2022.
Early official results were expected later on Sunday.
France's electoral system can make it hard to estimate the precise distribution of seats in the 577-seat National Assembly, and the final outcome will not be known until the end of the second round of voting on July 7.
Voting data suggests an unusually high turn out for France's first round elections. (EPA PHOTO)
"We are going to win an absolute majority," Le Pen said in a newspaper interview on Wednesday, predicting that her protege, 28-year-old Jordan Bardella, would be prime minister.
If the RN does win an absolute majority, French diplomacy could be headed for an unprecedented period of turbulence: with Macron - who has said he will continue his presidency until the end of his term in 2027 - and Bardella jostling for the right to speak for France.
France has had three periods of "cohabitation" - when the president and government are from opposite political camps - in its post-war history but none with such radically divergent world views competing at the top of the state.
Bardella says he would challenge Macron on global issues.
France could lurch from a pillar of the EU to a thorn in its side, demanding a rebate of its contribution to the EU budget, clashing with the bloc over European Commission jobs and reversing Macron's calls for greater EU unity on defence.
with AP
