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.