The head of the city’s Department of Cultural Affairs conceded Tuesday that major cultural institutions are still falling well short of meeting Mayor Bill de Blasio’s goals for fielding diverse workforces, but his agency has no immediate plans to go ahead with previous threats by yanking funding over it.

Tom Finkelpearl told a City Council hearing that the administration has instead opted to give Metropolitan Museum of Art, Lincoln Center, Carnegie Hall and other city-funded cultural institutions until next spring to prove they’re taking steps to become more diverse before deciding whether to pull some of their annual city funding.

Under a plan backed by de Blasio, art and cultural institutions with predominately white workers that refuse to change their ways risk losing up to 10 percent of city funding. These nonprofits now receive a total $117.2 million in taxpayer cash.

The commissioner said that while the city isn’t enforcing quotas it is waiting to see whether these institutions meet diversity “goals” they have promised the city they’d try reaching.

“The next threshold will be next spring,” Finkelpearl said “A lot of the organizations have set themselves some very specific kinds of goals. Those goals are different from organization to organization.”

Finkelpearl later clarified for reporters that the intuitions’ “goals” don’t necessarily mean having to reach specific numbers — but rather showing progress is being made in the area of workforce diversity.

“We are looking for good faith efforts,” he said.

Roughly two-thirds of staffers employed at city-funded museums, theaters, zoos and other institutions are white — even though white people comprise just 32 percent of the Big Apple’s population, according to a July report by SMU DataArts prepared at the request of City Hall.

“Clearly our work with the field is to cut out for us in terms of increasing racial diversity, particularly in senior level positions,” Finkelpearl said.

The survey of 6,928 workers at city-funded 65 art and cultural institutions did find that 15 percent of the arts workforce identify as gay, lesbian, bisexual or queer, including 25 percent in leadership positions.

“Gays love the arts, so it is important you include us,” said Queens Councilman Jimmy Van Bramer, who is gay, told Finkelpearl after the commissioner cited the report’s findings.

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