President Emmanuel Macron could fall short of a parliamentary majority after the French took to the polls in a first round of voting on Sunday.
Some polling companies see Mr Macron's centrist alliance scoring less than the 289 seats required.
Mr Macron's Ensemble alliance is said to be running neck-and-neck with a new left-wing alliance, Nupes, with both scoring about 25-26 per cent of the popular vote.
From these figures, polling firms projected that Ensemble would win 225 to 310 seats in the second round of voting next Sunday, possibly short of a majority.
Nupes, a newly unified alliance of leftists, Socialists, Greens and Communists, was forecast to win 150 to 220 seats, making it by far the biggest opposition force in Parliament.
If Mr Macron's coalition falls short of a majority, it is expected to lead to bill-by-bill deals with right-wing parties in Parliament, or he would have to try to poach opposition or independent MPs for his political group.
"It's a very serious warning that has been sent to Emmanuel Macron," political scientist Brice Teinturier told France 2 television.
"A majority is far from certain."
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Sunday's vote followed presidential elections in April in which Mr Macron secured a second term, beating far-right leader Marine Le Pen with pledges to cut taxes, reform welfare and raise the retirement age to 65 for most people.
After a dismal performance in that vote, the French left has united behind Jean-Luc Melenchon, a hard-left veteran who has a radically different programme, including lowering the retirement age, wealth taxes and increasing the minimum wage by 15 per cent.
Turnout was on course to be a record low of 47 to 47.5 per cent, polling projections showed.
"Some people say that parliamentary elections aren't important but that's not true," engineer Arnaud, 40, told AFP as he cast his vote in Paris.
"If the president doesn't win a majority he can't get anything done."
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Ms Le Pen's far-right National Rally was seen as winning 10 to 45 seats nationally, potentially sharply increasing the party's representation in Parliament from its current eight seats.
The record low turnout is set to confirm the trend of dwindling interest for parliamentary elections over the past 20 years.
"The very personal nature of the presidential election still continues to interest people, but it overshadows all the other types of ballot, even the essential one which is electing the members of the national assembly," said Mathieu Gallard of the Ipsos polling group.
While Mr Macron and his EU allies breathed a heavy sigh of relief after his solid presidential victory against Ms Le Pen, the past weeks have brought no sense of a honeymoon.
Energy and food prices are soaring in France as elsewhere in Europe, the treatment of English fans at the Champions League final in Paris damaged France's image abroad, and Mr Macron has been accused by Ukraine of being too accommodating with Russia.
His new Disabilities Minister Damien Abad has faced two rape accusations, which he has vehemently denied, while new Prime Minister Elisabeth Borne has yet to make an impact.
Mr Macron has made it clear that ministers who are standing in the election — including Ms Borne, who arrived on top on Sunday in her first attempt at winning a seat — will have to step down if they lose.
Europe Minister Clement Beaune, a close ally of Mr Macron and a crucial influence over France's Brexit and wider EU policies, is also standing in his first election and is considered to be in a close fight with a left-wing rival.
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Under France's system, a candidate needs more than half of the vote on the day, as well as the backing of at least 25 per cent of registered voters in a constituency to be elected outright in the first round.
Otherwise the top two candidates in a constituency, and any other candidate who won the backing of at least 12.5 per cent of registered voters, go forward to the second round, where the one with the most votes wins.
