The first thing Lucie Castets intends to do as France’s next prime minister is to peel back the age of retirement to 62. Then, she will pour more money into the creaking health care and education systems. To pay for at least some of that, she will introduce a tax on the country’s ultrarich.

There is just one hitch. Ms. Castets, the candidate of choice of the left-wing coalition that won the most seats in France’s snap legislative elections that ended in July, has not been tapped for the job. And the one person with the power to offer it to her, President Emmanuel Macron, has shown no sign he plans to do so.

“We are in a somewhat Kafkaesque, surreal situation, where a candidate for the post of prime minister is campaigning for a job that she cannot exercise,” Rémi Lefebvre, a professor of political science at the University of Lille, said.

Almost seven weeks since those elections ended in deadlock, with neither left, right nor center winning a majority, France remains intractably stuck.

Since then, Mr. Macron has been in no hurry to pick a new prime minister, whose job it is to run the country while the president officially oversees France’s institutions. He called for a political truce during the Olympics, which continued into the dog days of August, when the capital empties and anyone who can posts an “I’m on vacation” sign on the door and disappears.

On Friday, Mr. Macron began hosting a series of meetings with political leaders to help inform his choice. In the past, choosing was easier: He simply picked candidates from his own winning coalition and expected them to collaborate with him.

But since his coalition took a drubbing in the election and lost more than 80 seats in Parliament, that’s no longer a clear choice. Tradition would hold that he offer the choice to the winning party — but in the past, that party always had a majority, unlike now.

The president’s plan, his advisers have said, is to pick the person who could shore up the most political support and therefore run a stable government. His read of the election result, he told citizens in an open letter last month, was that voters were demanding a new, conciliatory style of politics with “power sharing” — something unknown to recent French political culture and that Mr. Macron himself pledged to introduce in 2022 but didn’t.

“In France, we are used to having either all the power or none,” a left-wing politician, Raphaël Glucksmann, explained to Le Point magazine.

That sensibility has left nearly everyone talking about compromise but offering little of it. The leaders of the four left-wing parties that banded together to form a coalition called the New Popular Front say someone from their group should be given the job, since they won the most seats in the election. But many of the actions the coalition has vowed to champion run counter to Mr. Macron’s philosophy of making France more business-friendly, including trying to overturn what he considers the hard-fought success of raising the retirement age.

After arguing for two weeks over whom they would put forward, they settled on Ms. Castets — plucking her from the relative political obscurity of her office at Paris City Hall, where until then she had run the financial department.

The request, which came over the phone while Ms. Castets was biking, shocked even her. She has never run for, let alone held, public office. And while she is a vocal supporter of left-wing causes — particularly in support of France’s vaunted public service — she left the Socialist Party years ago. Since then, she has not been a member of any party.

“They know people want some fresh air,” said Ms. Castets, 37, during an interview this week, squeezed in between meetings. “They want people who were not involved in the parties — they want something else.”

She added, “I don’t want to be president in 2027. I’m not a threat to them.”

Still, this lifelong civil servant has been doing something political analysts say they’ve never seen before: campaigning for the job of prime minister.

She has been introducing herself to fellow citizens, in person and through the media, as a lesbian married mother of a 2-year-old, a graduate of the country’s top college for civil servants who loves to be physically active. She has been fielding interviews. She has sent letters to parliamentarians, co-signed by leaders of the parties, laying out her potential government’s plans.

Over the weeks, she has softened her position from its original hard-core stance that her government would follow through on the entire left-wing coalition program. Now, she says she would pursue something more reflective of minority government position.

Her government would focus on goals she believes she could find wider agreement on, she says, among them increasing the minimum wage, making the economy more environmentally sound and investing in public services.

“Who will accept the continued catastrophic situation in public hospitals with emergency services closed in the middle of summer?” she said in an open letter this week to fellow citizens, co-signed by political backers who all vowed to invent a new way of governing. “Who will settle for a new school year where there will be a shortage of teachers facing our children in elementary, middle and high schools?”

Still, she is facing an uphill battle, and not only because she is unknown and politically inexperienced. The biggest party in her coalition, France Unbowed, has a history of scorched-earth politics that makes the pledge for conciliation feel thin.

On Thursday, Aurore Bergé, a departing minister and leading member of Macron’s party, Renaissance, repeated on the radio that she would vote immediately to bring down any government with far-left France Unbowed members in it. She and many others consider France Unbowed and its combative leader Jean-Luc Mélenchon, a former Trotskyist, to be as dangerous to France’s democracy as the extreme right.

Ms. Castets is not the only one openly vying to be part of a new government. The prime minister, Gabriel Attal, and foreign minister, Stephane Séjourné, both historically key players in Renaissance and part of the current caretaker government, sent an “action pact for the French” to fellow politicians, laying out seven areas they believed offered ground for collaboration.

The leaders of the country’s moderate Conservatives, now called the Republican Right, also published an “emergency legislative pact.”

If the president does not pick someone from the left or someone from center who could corral more votes than New Popular Front, the message to citizens would be scathing, said Bruno Cautrès, a political scientist at the Center for Political Research at Sciences Po university in Paris: “that Emmanuel Macron did what he wanted and didn’t take into account the election results.”

While the constitution does not lay down rules on whom the president should name as prime minister or by when, a kind of deadline is looming. Normally, the government must introduce its budget by early October so Parliament can pass it by the end of the year.

“It would be catastrophic for Emmanuel Macron if by Jan. 1, 2025, the French Parliament was unable to adopt a budget,” Mr. Cautrès said. He noted that the state could still function without one but that France’s image would take a serious bruising in the international financial markets.

To get that all done, a government would need to be in place by the end of August or the first days of September, he said.

Mr. Macron’s team said he would announce his choice “quite soon.”

To many, the road ahead looks bleak, and whoever takes the job seems destined to fail. Right-wing lawmakers have said they will immediately bring down any left-wing government. France Unbowed lawmakers have said they will censure any prime minister who is not Ms. Castets.

“The crisis will deepen,” Mr. Lefebvre said. “I don’t see any viable institutional or political solution emerging over the coming weeks and months.”