Such an outcome appeared uncertain before Macron's presidential victory a month ago, with many political pundits skeptical about whether the former banker's upstart party could secure over half the 577 seats in the National Assembly with many candidates who are virtually unknown to voters.
But now even some officials in former President Francois Hollande's Socialist party and the far-right National Front (FN) are publicly saying LREM is likely to win a majority and are simply urging voters to make the margin as small as possible.
Meanwhile conservative grouping the Republicans, which polls in second place after LREM, is barely clinging to fading hopes it can hold the balance of power.
"It's all exploding. This was a solid fiefdom for us, but our huge pool of voters dried up after the presidential election," Sebastien Vincini, a top Socialist official, said of his south-western Occitanie region, where Socialists held two thirds of the constituencies but expect to lose most of them.
FN chief Marine Le Pen topped the first round of the presidential election in April in Vincini's Occitanie region, and Jean-Luc Melenchon's far-left France Unbowed, which helped kill the Socialists' chances in the presidentials, is getting stronger there.
But still for Vincini, "the biggest threat ... comes from French voters' desire to give Macron a majority".
According to pollsters, that is true nationwide ahead of the parliamentary ballots.
A weekly OpinonWay survey sees LREM getting 335-355 seats out of 577, with 145-165 for The Republicans, a humiliating 20-35 seats for the outgoing Socialist majority, and 21-34 for Melenchon and his communist allies.
Macron and his team are keenly aware of the importance of a majority to his reforms plans, which include overhauling labor laws to favor business, cutting corporate tax, and investing 50 billion euros of public cash over five years in areas including job training and renewable energy.
"If we don't have a majority we'll be stuck in an in-between place," 28-year-old Pierre Person, an adviser to Macron and a parliamentary candidate, told Reuters as he handed out leaflets at a bustling Paris food market.
Much is also at stake for Le Pen and her deputy Florian Philippot. A disappointing score for the FN could spell trouble for their leadership, which has already been weakened by her lower-than-expected share of the votes in the presidential election run-off against Macron.
The prospect of local anti-FN alliances under which candidates agree to pull out and back its strongest rival means the far-right party may be the worst represented of all the main groups, despite polls which show that between a quarter and a third of voters back its policies.
There are a total of 7,882 candidates running for 577 constituencies. If, in any one seat, no candidate wins outright with 50 percent of the vote on Sunday, all those with at least 12.5 percent qualify for the June 18 run-off.
Among LREM candidates challenging traditional strongholds of other parties is Mounir Mahjoubi, a 33-year-old junior minister who is facing Socialist Party chief Jean-Christophe Cambadelis in a Paris constituency where the 65-year-old has been lawmaker for 20 straight years.
Other LREM candidates include a former bullfighter, a mathematician, pensioners and business owners, many of whom have no political experience.
Even if French polls are proven right and Macron wins that solid majority, he will face the challenge of keeping such a diverse group united.
Some of the people in his team are Socialist rebels, while others are from the center-leaning camp of The Republicans. The ideological gap between those two is a wide one.
Nothing is certain, however, despite the pollsters' predictions. Sleaze allegations against a Macron minister, a likely low turnout and the complex permutations of the run-off system could still throw up surprise results.
Macron's presidential victory, his party's meteoric rise and a near faultless first few weeks in power are nonetheless forcing parliamentary candidates to re-think their electoral strategy.
Many who are not running under LREM's colors still mention on posters and leaflets that they are candidates for the "presidential majority" - meaning they will back laws that the president puts to parliament - or at least say they will endorse Macron if elected.
These include some Socialist or Republican candidates, and even some who are running against LREM candidates. One Socialist candidate even included a picture of himself shaking Macron's hand on his campaign poster.
SEEDS OF DIVISION
These ambiguous positions, and the radically new political order they engender, are making life difficult for Republican and Socialist loyalists who hope to rebuild their shattered parties.
While The Republicans are expected to be the biggest opposition party, the seeds of division have been sown.
The prime minister and two of Macron's ministers, one of whom is running for parliament under the LREM label, come from its own ranks; some Republican candidates say they would vote a number of Macron's reforms, while others reject any kind of cooperation.
To the far-left of the political spectrum, Melenchon, a fiery orator, is setting himself up as the future of the French Left - making life tougher still for the Socialists who has also lost some members to the Macron cause.
Melenchon is a candidate in the southern city of Marseille, where a local poll shows he has good chances of winning.
"I don't want to weaken the Socialist party, I want to replace it," Melenchon told reporters. He attracted a fifth of votes nationally in the first round of the presidentials.
If elected, even with a small parliamentary group, Melenchon would likely be one of Macron's most vocal opponents in the assembly alongside Le Pen, also a failed presidential candidate, who is seen as likely to win the northern France constituency of Henin-Beaumont.
The FN and its far-right allies only won two seats in 2012, but has been targeting as many as 40 deputies this time around.
But while a seat in parliament would give Le Pen a platform to challenge Macron, polls show it is touch and go whether she will reach 15-seat threshold to form a parliamentary group that delivers more speaking time and access to top roles within the assembly.
"The harvest will be of average quality," an FN official told Reuters on condition of anonymity.
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