The FN leader is now a lawmaker in the European Parliament but is hoping to win a legislative seat in France for the first time.
"It is neither healthy nor desirable for a president who gathered only 24 percent of the vote in the first round of the presidentials and who was elected in the second round only by the rejection of the extreme right should benefit from a monopoly of national representation", said Socialist Party leader Jean-Christophe Cambadelis as results flowed in.
If no candidate manages to achieve that target then all candidates who win at least 12.5% of registered voters will go into the second round, where the victor will advance to parliament.
Macron's camp was comfortably leading with more than 32 percent - putting it well ahead of all opponents going into the decisive second round of voting next Sunday for the 577 seats in the lower-house National Assembly.More news: Xbox One X Formally Unveiled, Coming This Fall
Mounir Mahjoubi, junior minister in charge of digital affairs, said on BFM television that voters have acknowledged that the first weeks of Mr Macron's presidency "have been exemplary" and "have allowed the French to see there is a path that suits them".
The head of the conservative Republicans party, Francois Baroin, also urged voters to turn out in larger numbers next week to help ensure that Mr Macron's party faces a robust opposition.
Only 48.8 percent of registered voters cast their ballot on Sunday, the lowest turnout in French modern politics.
Benoit Hamon, the party's presidential candidate, was also knocked out of the running for his seat.
The National Front of far-right leader Marine Le Pen looked unlikely to convert her strong showing in the presidential election into a large number of legislative seats.
Macron's office says he has held a series of conversations over the past week with the emir of Qatar, the king of Saudi Arabia, the Turkish president and the crown prince of Abu Dhabi.
Sunday's results suggest that Macron's party could end up with between 415 and 455 seats out of 577 after next week's run-off.More news: Reports swirl about Beyonce's twins, but no confirmation yet
That was Le Pen's hope after she advanced for the first time to the presidential runoff that Macron won on May 7.
The Republicans are projected to have taken 20.9% of the vote, which is predicted to result in the party winning between 80-100 seats.
Opinion polls showed Macrons wouldwin between 360 and 427 seats - easily a majority.
FN vice-president Florian Philippot admitted to "disappointment" and called on voters to "mobilise massively" for the June 18 second round.
- "Monochrome parliament" -But former prime minister Alain Juppe of the rightwing Republicans said the low turnout was a sign of "deep malaise" in the electorate and that a clean sweep by Macron would be bad for democracy.
If the seat projections are confirmed next week, he will have a strong mandate to push through the ambitious labour, economic and social reforms he promised on the campaign trail.More news: Trump tweet confirms he's being probed by prosecutor
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