“I’m a big fan of Russian oligarchs paying more to get into the Met,” he said.

Yet the possibility of distinguishing among visitors — and even of making the admission “suggested” for city residents but required for state residents — risks alienating tourists and New Yorkers alike, especially if people see the fee as a way for the Met to solve financial and governance missteps that have led to a budget deficit of about $15 million. Met leaders have resorted to cost-cutting moves, including trimming staff and reducing the number of annual exhibitions to about 40, from nearly 60; a fee would establish the kind of large and consistent revenue stream that private museums already enjoy.

Unlike several Manhattan museums that charge admission, the Met is considered a public museum: New York City owns its building and also gives the museum $26 million a year — the largest amount that the city provides to an arts institution. Still, the city’s appropriation to the Met covers just about 8 percent of the museum’s $332 million annual operating costs.

The Met’s current “suggested” adult admissions fee, $25, generated about $39 million in the fiscal year 2016, or 13 percent of the museum’s overall revenue. A mandatory fee would be likely to generate tens of millions of dollars more a year, providing a much-needed shot in the arm.

Should the Met earn these additional millions, the city might become more inclined to shift public money to other arts groups, a prospect that city officials are now considering as part of a new Cultural Plan, to be released soon. The city may view the Met’s new income as a replacement for its public dollars, which could be risky if admission fees prove less reliable than annual city funding.

But the mayor said Wednesday that an admission fee would not necessarily jeopardize city support.

“I don’t think we have an assumption about city funding,” Mr. de Blasio said. “It’s about them being able to sustain their operations long term and would actually mean they wouldn’t need additional city funding, in theory. But no, the real issue is just to allow them to defray some of their costs in a way that’s fair to city residents.”