The Met is close to a deal with the city under which the museum would accept a reduced annual allocation in exchange for the city’s approval of the Met’s plan to make up that revenue by charging $25 admission for visitors from outside New York State, according to a cultural executive in New York museum circles with knowledge of the negotiations, who spoke on condition of anonymity because the person was not authorized by the Met or the city to comment. (A Met spokesman would say only that the museum was still in discussions with the city about the proposed admission fee.)

Moreover, Mr. de Blasio is suddenly brandishing a stick after years of rarely visiting or championing cultural institutions, in contrast with his predecessor, Michael R. Bloomberg. When asked directly at Wednesday’s news conference whether he’d ever been to MoMA/P.S. 1, a nearby museum, Mr. de Blasio answered simply, “No.”

Several board leaders of major museums did not immediately respond to requests for comment. Yet some city officials were quick to go on the offensive against any potential opposition.

“There are people who will resist, who will resent, who will obstruct,” said Melissa Mark-Viverito, the City Council speaker. “Any moment for equality and inclusion doesn’t come easily.”

For now, the plan calls for maintaining city funding for New York’s most popular museums, allaying concerns that some of their subsidies would be redirected to smaller arts organizations. Mr. de Blasio pledged on Wednesday to increase money for the less prominent groups, many in neighborhoods outside Manhattan.

Mr. Finkelpearl acknowledged in an interview that the cultural plan was a road map rather than a guarantee of future changes to city arts funding. “This is not a budget document,” he said. “This is a document that sets a course. All this stuff will be taken into consideration in future-year budgets.”