Marine Le Pen's far-right National Rally (RN) party scored historic gains to win the first round of France's parliamentary election, but the final outcome will depend on days of alliance-building before next week's run-off vote.

The RN and allies had 33% of the vote, followed by a leftwing bloc with 28% and President Emmanuel Macron's centrists with just 20%, official results from the interior ministry showed on Monday.

That was a huge setback for Macron who had called the snap election after his ticket was trounced by the RN in European Parliament elections last month.

But whether the anti-immigrant, eurosceptic RN will be able to form a government will depend on next week's decisive round and how successfully other parties manage to thwart Le Pen by rallying round the best-placed rival candidates in constituencies across France.

Leaders of both the leftwing New Popular Front and Macron's centrist alliance made clear on Sunday night they would withdraw their own candidates in districts where another candidate was better placed to beat the RN in next Sunday's runoff.

A longtime pariah for many in France, the RN is now closer to power than it has ever been. Le Pen has sought to clean up the image of a party known for racism and antisemitism, a tactic that has worked amid voter anger at Macron, the high cost of living and growing concerns over immigration.

An RN-led government would raise major questions over where the European Union was headed, given its resistance to further EU integration. Economists have also asked whether its spending plans are fully funded.

The euro touched a two-week high during Asian trading on Monday on market relief the RN had not done better.

"I think it's a slight 'well, there were no surprises', so there was a sense of relief there," said Fiona Cincotta, senior markets analyst at City Index.

RN lawmakers on Monday urged centre-right politicians in the Republicans (LR) party, which received less than 7% of the first-round vote, to withdraw from districts where such a move would work in RN's favour.

"If they know they're not going to win, I'm calling on them to stand down and let the national side win," RN lawmaker Laure Lavalette told RTL radio.

For now, the Republicans party, which split ahead of the vote with a small number of its lawmakers joining the RN, has given no indication of its stance.

All candidates who made it through the first round have until Tuesday evening to confirm whether they will go into the second.