Nearly half the city’s prison population has been diagnosed with mental health problems, but first lady Chirlane McCray’s $250 million-a-year “ThriveNYC” mental health initiative has no presence in the jails.

“Thrive does not work in the facilities,” Correction Commissioner Cynthia Brann testified at a city council budget hearing Thursday.

“We are engaged with the first lady’s Thrive initiative. We are aware of it, but we don’t have staff from Thrive in jails.”

Brann said all inmates’ medical needs — including mental health — are served by Correctional Health Services, a branch of the municipal hospital system that operates with its own $253 million budget.

About 43 percent of the city’s 8,000 inmates have been diagnosed with mental health problems.

Queens Councilman Robert Holden said Brann’s comments raise “serious red flags.”

“You’d expect [Thrive] to be in the jails giving direct help … considering mental illness is one of the biggest problems [the city’s prison system] is facing,” he said.

“We need to start seeing some results from Thrive here, and the results need to be visible.”

Raul Contreras, a City Hall spokesman, said Thrive provides “support services” to jails even if it doesn’t staff the facilities.

“While it’s true that Thrive does not staff city jails…[it] funds therapeutic creative arts programming, psychiatric assessments and substance use prevention for all young adults currently housed on Rikers Island… Thrive also has successfully trained 7,000 correction officers in Mental Health First Aid,” he said.

In a related development, McCray announced Thursday said she would accept Council Speaker Corey Johnson’s invitation to testify at a March 26 hearing on Thrive and answer critics who question whether the program is wasting money.

“I look forward to dispelling that during the hearing,” said McCray, who was in Washington, DC, to deliver remarks at a mental health forum.

Asked if she had the numbers to back that up, she responded: “Of course, of course.”

The number of complaints to the NYPD about emotionally disturbed individuals jumped 23 percent since McCray launched her program.

Cops received 179,569 calls of people in mental distress in 2018, up from 145,430 in 2015 — the year that Thrive NYC launched, according to New York 1.

Data from the US Department of Housing and Urban Development shows the number of homeless people in New York City with serious mental health issues also spiked from 9,840 in 2015 to 12,140 by 2018.

Additional reporting by Nikki Schwab