From political circles in New York City to cocktail parties on Capitol Hill, on right-of-center Facebook pages and among left-of-center donors, two of the biggest untethered threads in New York politics are being drawn together around a single question.

Would Hillary Clinton run for mayor?

The prospect has an obvious, novelistic allure: A run for mayor of New York this year would pit Mrs. Clinton against Mayor Bill de Blasio, a fellow Democrat who managed her Senate campaign in 2000, and, should she win, would put her in charge of President-elect Donald J. Trump’s hometown, ensuring years of potential clashes between bitter rivals.

The question has gained momentum in the weeks since Mrs. Clinton lost to Mr. Trump, a Republican. It began quietly at first, then grew louder among the conservative media outlets and pundits who hold her and Mr. de Blasio in disdain.

While the answer would almost certainly be no, supporters of Mrs. Clinton and former members of her staff appear to have been happy to let the speculation spread from closed-door gatherings of donors and allies, where it has been discussed among the many hypothetical future jobs Ms. Clinton might pursue, to more public forums.