• After the left wing's election success in France, all eyes are now on radical firebrand Jean-Luc Mélenchon who has demanded the left be given the premiership.
• "The president has the power, the president has the duty to call on the New Popular Front to govern," Mélenchon said Sunday night.
• The expansionary fiscal policy of the left-wing New Popular Front has been a cause of concern for economists since Macron called the snap election last month.

After the left wing's election success in France on Sunday, all eyes are now on radical firebrand Jean-Luc Mélenchon, who has demanded the left be given the premiership and a chance to govern following its unexpected election win.

The New Popular Front (NFP) coalition — of which Mélenchon is the self-appointed figurehead — won the largest number of seats in the second round of France's snap parliamentary election. Comprising Mélenchon's far-left France Unbowed party, the Socialist Party, the French Communist Party as well as green, center-left and left-wing political groups, the NFP unexpectedly thwarted the far right's advance and is now positioning itself as the possible leader of a coalition government.

"The president has the power, the president has the duty to call on the New Popular Front to govern. It is ready for it," Mélenchon said Sunday night after exit polls projected the NFP's win.

Europe had braced itself for France's far right to win the largest number of votes in the second round of the country's snap election. In the event, the left-wing NFP gained 180 seats, according to results published by broadcaster France Info, but still fell short of the 289 seats needed to have an absolute majority in the 577-seat National Assembly. French President Emmanuel Macron's centrist "Together" bloc came in second with 163 seats and the far-right National Rally and its allies won 143 seats.

The results mean France is confronting a hung parliament Monday morning, with a difficult path ahead to forming a new government, or perhaps a technocratic government.

Prime Minister Gabriel Attal said he would tender his resignation following the election result, but Macron on Monday asked Attal to remain prime minister for the time being "in order to ensure the country's stability," Macron's office said in a statement, Reuters reported.

Emboldened by the unexpected election victory, Mélenchon — who in the past has praised late Venezuelan dictator Hugo Chavez, said he'd take France out of NATO and proposed 100% tax rates on France's super wealthy — insisted the new prime minister should come from the left-wing alliance.

Political analysts and economists say Macron is highly unlikely to give the divisive figure of Mélenchon the job.

"While tradition dictates that the largest party in parliament (in this case, the left-wing NFP coalition) proposes a prime minister, Macron is not obligated to pursue such an option," Antonio Barroso, deputy director of research at consultancy Teneo, said in a note Sunday.

"Mélenchon has said the new prime minister should hail from NFP. However, the NFP has no leader, and the parties forming the coalition are strongly divided over whom they could select."

"Even in a scenario where the NFP were to agree on a figure and Macron appointed such an individual, the AN [National Assembly] could then easily force a collapse of the new government via a no-confidence motion," Barroso added.

Whoever is proposed as the new prime minister will have to command the support of centrist parties in the lower chamber of parliament in order to reach an absolute majority, he added. "As a result, the appointment of a radical figure such as Mélenchon is off the table."

Economists expect a centrist

The expansionary fiscal policy of the left-wing New Popular Front has been a cause of concern for economists since Macron called the snap election last month, after his party's defeat in European Parliament elections in early June.

France is already facing a challenging fiscal position and the European Commission announced in June that it intended to place France under an excessive deficit procedure due to its failure to keep its budget deficit within 3% of GDP. If the EDP is approved, France will have to present amended budget plans by mid-September.

"Mélenchon is not as well-known outside of French politics as [the far-right National Rally's] Marine Le Pen, but he and the other leaders of the leftist alliance have proposed a program that includes big increases in public spending, rolling back the retirement age and other policies that are at odds with the EU — and are estimated to cost an additional 179 billion euro [$194 billion], according to the Institut Montaigne. Market-friendly they are not," Tina Fordham, founder of Fordham Global Insight, said in a note.

Armin Steinbach, law and economics professor at HEC Paris, said the NFP would likely opt for a prime ministerial candidate with broader appeal, such as former President Francois Hollande, who was in power from 2012-2017, prior to Macron.

"The left bloc must come up with a proposal of a moderate politician so it can't be someone from La France Insoumise [France Unbowed], it has to be someone with a moderate position, so someone like [former Socialist President] Francois Hollande, he has institutional knowledge so he could be one potential candidate."

Ludovic Subran, chief economist at Allianz, also believed centrists within the NFP alliance would come to the fore in forthcoming coalition talks, and that Hollande could be a possible candidate for PM.

"I'm not too worried, I think we are really converging toward a center-left policy building and the last time France adjusted its deficit was when Francois Hollande was president, so things can happen under a center-left government that one could see as positive for France in Europe," he told CNBC's Charlotte Reed.