Candidates from NUPES have accused the French Interior Ministry of manipulating the first-round results.
Four days before France heads to the polls for the second round of its legislative elections, questions have been raised about the official results so far.
Members of the NUPES leftist bloc of Jean-Luc Mélenchon have accused the authorities of omitting some of their votes and mislabelling certain candidates.
The gap between Macron's centre-right Ensemble! alliance and the left-wing group in the first round was just 21,359 votes.
According to the French Interior Ministry, Ensemble! picked up 25.8% of the vote, narrowly ahead of NUPES' 25.7%.
But members of the left-wing alliance have accused the French government of manipulating the results.
Manuel Bompard, a senior NUPES politician, has claimed on Twitter that 265,000 left-wing alliance votes were not counted in the final totals.
French media -- including Le Monde -- have also placed NUPES ahead of Ensemble! with 26.1% of the vote.
Former French Interior Minister Christophe Castaner has dismissed the claims by NUPES as a "classic conspiracy theory".
Where does the discrepancy in numbers come from?
Bompard and other NUPES candidates have claimed that certain left-wing members have not been included in the official tally.
This includes Dominique Potier, Joël Arivagnet and Hervé Saulignac, as well as other left-wing candidates in French overseas territories.
Potier, Arivagnet, and Saulignac all ran against Macron's Ensemble! alliance in their respective constituencies.
But while they were initially registered as members of NUPES, they withdrew from the alliance before Sunday's election and ran as independent left-wing candidates.
"I stand as a social-democrat, ecologist and European candidate," Potier told France3 in an interview last month.
"I have decided not to wear the colours of this agreement of popular union and to place myself in the perspective of the construction of a new left."
The French Interior Ministry says it was provided with a list of all NUPES members before the vote to use in their official results publications. Potier, Arivagnet, and Saulignac were not on that original list and the latter two were later added to the NUPES website on Monday, after the vote.
Moreover, left-wing candidates in overseas territories such as Corsica and Guadeloupe were not included on the alliance list due to the individualities of their local parties and all ran separately.
The controversy has generated fresh discussion about the way France counts votes during its legislative elections.
Just six days before the first round, the country's Council of State had stated that NUPES votes should be counted as a single political party.
The French government had argued that each individual left-wing party should run separately, but the council said in a statement that this may "undermine the fairness of the presentation of the election results".
Unfounded allegations of voter fraud had previously spread on social media following France's presidential election in April.