Long Island and New York City officials said Monday in response to an order from Gov. Andrew M. Cuomo that they will still make case-by-case decisions on whether to force homeless persons into shelter on the coldest winter nights.

Cuomo administration counsel Alphonso David said that the executive order the governor issued Sunday didn’t take away such discretion.

Cuomo and local governments agree that those deemed mentally ill can be forced off streets for evaluation and treatment, as was already the case. Some local officials, including aides to New York City Mayor Bill de Blasio, had raised concerns that wider sweeps would result in legal challenges.

The Democratic governor Monday turned his focus to “unacceptable” conditions at shelters, saying that many of the homeless prefer the frigid outdoors over facilities that are dirty and dangerous.

“We can’t have people staying on the streets because the streets are better than the shelters,” Cuomo said after an event in Manhattan. Promising to announce reforms in his Jan. 13 State of the State address, he said, “You need a shelter system that has the capacity and is safe, clean, decent and habitable so the homeless will go there.”

Nassau County police will bring in only the homeless who voluntarily accept an offer of shelter, county spokesman Brian Nevin said. Those deemed mentally incompetent will be taken to a hospital — the county’s current procedure, he said.

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Suffolk Deputy County Executive Jon Schneider also said police will not take people off the streets against their will unless it is determined that they are incompetent. Schneider said the governor’s office conducted a conference call Monday with the department of social services officials to clarify the order, which applies when temperatures are below 32 degrees.

“Basically, what Suffolk County already does falls within what the state is looking for this year,” he said.

De Blasio, who has clashed with Cuomo on the city’s homelessness crisis, said at a news conference that the city was already fulfilling the aims of Cuomo’s order and pointed to city initiatives including HomeStat outreach teams.

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“It seems to simply reiterate what’s already in the law, and the power we already have to bring people in off the streets,” he said in Brooklyn. “And we use that power. Let’s be clear.”

City Coalition for the Homeless president Mary Brosnahan said Cuomo’s administration “has started to backtrack” on an initial edict that had appeared to call on local officials to “forcibly move individuals against their will.”

David said that while competent individuals cannot be moved against their will, the great majority of those sleeping outside on freezing nights likely suffer from mental illness.

The state will monitor compliance with the order through its Office of Temporary and Disability Assistance, Cuomo spokeswoman Dani Lever said.

In lower Manhattan, Amy Kaufman, 43, of Detroit, and her partner, Luan Barcelos, 27, of Brazil, said shelter workers Sunday night tried to get them off the street, but she refused.

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“We said no. It is filthy. The people there steal from you and you get into fights and the cops are called and that’s no good,” said Kaufman.

The couple bundles up at night in a building doorway sheltered from the wind. “I prefer to be outside in the 5 degree weather. . . . It’s not easy. But it’s freedom.’’

With Robert Brodsky, David Schwartz, Maria Alvarez and Alison Fox