A group of Manhattan residents is demanding that the city Department of Investigation probe a $500 million, 40-year government contract involving a shelter for women with drug addiction and mental illness.

The contract for the 200-bed shelter at 37 W. 59th St. — next to Ederle Playground and a few blocks south of Lincoln Center — was awarded during the final months of the de Blasio administration.

“The community rightfully believes that where there is smoke, there is fire and entrusts that the DOI will investigate this exorbitant contract approval at the 11th hour of the previous administration with the seriousness it deserves,” wrote Brad Gerstman, a lawyer who represents the Friends of Ederle Playground group, in a letter sent this month to the DOI and obtained by The Post.

The community is “extremely uneasy about the timing as well as the players involved and the circumstances surrounding the approval of this $500 million taxpayer funded contract,’’ Gerstman added in his missive to John Bellanie, the DOI’s inspector general for the Department of Homeless Services.

The city DHS’s $500 million contract is with Project Renewal and runs from August 2021 through June 2060. The developer is The Hudson Companies.

The shelter site, also located across from CUNY’s John Jay College, was the subject of a parents’ protest rally against it last month.

The Post reported in 2016 that Hudson Companies won the right to transform a Brooklyn library into a residential skyscraper, despite being outbid by at least two other builders.

Hudson CEO David Kramer was a mayoral friend and fund-raiser.

Gerstman’s letter noted the issue to DOI officials.

He also pointed out that Doreen Thomann-Howe, DHS’s former deputy commissioner of family services, is now Project Renewal’s chief operating officer.

She was hired in October and has worked at DHS for seven years.

The shelter, currently under construction, will be nine stories high.

Gerstman claimed the $500 million price tag was the largest contract approved by the city in the 2022 fiscal year, which city officials defended as fair and reasonable “without any further justification.”

A Post visit Sunday found both opponents and backers of the project.

Heather Groeger, 42, a nurse anesthetist with a 7-year-old son, said the shelter is not suitable for the residential “family friendly and child-centric” neighborhood — particularly next to the popular Ederle playground.

“It’s going to be bringing a large number of people seeking opioid addiction treatment into a residential neighborhood. And they’re not planning to have a waiting room,’’ she said.

But some other residents interviewed said the women’s shelter is a good thing.

“I think that the benefits of providing help for people in this city that need it would outweigh the risk of any potential crime that could come with them,’’ said Eneuri Acosta, 41, who lives a few blocks away.

“In fact, I would make the argument that the opposite would happen, that the longer that we go without providing treatment and help for those in need, that will potentially cause more problems.’’

The city’s shelter system is now bursting with a record-breaking 104,000 residents, more than half of them migrants.

West Side Councilwoman Gale Brewer, the former Manhattan borough president, said she opposes the project in favor of permanent housing.

But she said city officials are standing by the project and refusing to make any changes.

“My goodness sake. Can’t we make it permanent housing? Why are we doing another shelter!” said Brewer, who has written a series of letters to city officials opposing the project.

DOI declined comment regarding the letter.

DHS had no immediate comment, but a spokesperson previously defended the project.

“We are committed to ensuring that every community has the critical social safety net resources to help their neighbors in need,” the rep said.

“This high-quality shelter will be the first-of-its-kind in this community offering women experiencing homelessness the critical opportunity to receive the support they need and deserve to get back on their feet.”