The heads of the Metropolitan Museum of Art, Lincoln Center and Carnegie Hall are among dozens of cultural institution leaders quaking in their boots as Mayor Bill de Blasio considers slashing their funds if they fail to meet his staff diversity criteria, The Post has learned.

“We are not treating diversity as a bonus,” said Ryan Max, a spokesman for the Department of Cultural Affairs.

“These institutions won’t receive their full funding without demonstrating their commitment to equity and inclusion.”

Cultural Affairs officials are reviewing the plans that the mayor ordered drawn up by the landmark attractions — with potentially millions of dollars at stake if they don’t pass muster.

De Blasio’s decree targets the nonprofit organizations that run the 33 museums, theaters, concert halls, botanical gardens and zoos that comprise the city’s “Cultural Institutions Group.”

In addition to being located on city-owned land, each institution receives taxpayer money to fund its operations and subsidize energy costs.

The nonprofits now receive a total $117.2 million in taxpayer cash, and de Blasio has already proposed trimming $1 million from that amount as part of his budget cuts for fiscal 2020, which begins July 1.

It’s unclear how much more money could be withheld as punishment for failing to comply with Hizzoner’s affirmative action directive.

When Cultural Affairs Commissioner Thomas Finkelpearl first floated the idea in 2015, he insisted that City Hall wasn’t “looking to be punitive.”

But in 2017, de Blasio reversed course and warned that diversity “will be a factor in funding decisions,” claiming, “We do this because we believe in fairness.”

As part of the mayor’s “CreateNYC” program, the city sent consultants to help institutions revamp their recruitment, hiring and promotion practices.

The move is intended to help boost people with African, Latin, Asian, Arab and Native American heritage, “LGBTIQ populations,” people with disabilities, non-English speakers and the poor.

The consultants were paid in part with a $300,000 grant from the Ford Foundation, whose president, Darren Walker, said he was unaware that the plans could result in financial penalties.

The city also distributed an “Evaluation Framework for Diversity, Equity, Inclusion Plans” that will be used to grade each institution’s “goals,” “indicators” and “accountability.”

But the seven-page document doesn’t set any standards by which to define success.

The submissions were due April 30, and the Department of Cultural Affairs will review them

over the next six to eight weeks, with funding decisions to follow, Max said.

Max wouldn’t say how the plans will be scored, but said the city won’t impose quotas because they’re illegal.

City Councilman Joe Borelli (R-Staten Island) blasted the mayor’s plan as vague and potentially discriminatory.

“I can’t fathom how we can use diversity as a criterion without being able to define parameters, but then again, I can’t see how we can tie public funds to the ethnicity of an organization’s management,” he said.

Borelli also questioned why de Blasio hasn’t “led by example” and replaced Finkelpearl — a white male — “with a person of color.”

Additional reporting by Bruce Golding