During President Trump’s first months in office, officials in his administration have used the threat of withholding federal funding to bend institutions and municipalities to their ideological will.

That strategy shows no signs of abating.

The Department of Homeland Security is reviewing billions of dollars in grants for cities and states to make sure recipients comply with Mr. Trump’s priorities on immigration enforcement and diversity programs.

Now grant beneficiaries must “honor requests for cooperation, such as participation in joint operations, sharing of information or requests for short-term detention of an alien pursuant to a valid detainer,” according to the terms and conditions for grants distributed by the department.

The terms and conditions, which were first released last October and quietly updated in late March, also mandate that those who receive grants “will provide access to detainees, such as when an immigration officer seeks to interview a person who might be a removable alien.”

“I see it as the beginning of the reshaping of our national emergency management infrastructure,” said Mark Ghilarducci, who previously served as California’s state emergency services director. “Now it will be based upon ideology versus what is the actual need.”

The changes likely imperil tens of billions of dollars in grants to states and cities that have rules limiting cooperation with officials from U.S. Immigration and Customs Enforcement, he said.

The review of the federal grants is the latest reminder that Mr. Trump does not view federal funding to bolster police departments or help communities recover from disasters as support that comes without strings attached. Instead, he uses federal grants as leverage against state and local governments to ensure compliance with his political agenda.

One criterion is whether certain locales are sanctuary jurisdictions, which broadly refers to cities and counties that block their local jails from cooperating with federal immigration officials.

ICE prefers to pick up undocumented immigrants from local lockups, but in order to do so, it needs collaboration from county sheriffs. In some cities and counties, this collaboration is outright blocked or severely limited.

Peter Gaynor, who served as FEMA administrator and briefly as acting U.S. secretary of homeland security during Mr. Trump’s first term, saw these changes as in line with the priorities the president has set out.

“This is the way they are going to operate,” Mr. Gaynor said. “The administration is going to, from time to time, check you to make sure that you’re compliant with the federal grant guidance.”

“You don’t have to accept the money,” he added. “You can decline the money. I think that’s part of the formula here.”

California and some of its cities and counties have laws against cooperating with ICE. Now local governments and community organizations are at risk of losing billions of dollars because of the change in D.H.S. policy, said Brian Ferguson, a spokesman for the state’s governor, Gavin Newsom.

The department normally provides grants that support a variety of public safety services, including police, fire and emergency response as well as programs to prevent domestic violence and sexual assault.

“These are grants that are critical to prepare for the next disaster and help keep communities safe,” Mr. Ferguson said.

It is unknown exactly how much money is at risk in California or which cities, counties or community groups may lose grants, he said.

“Like much of the work that’s happening at the federal level, we’re working to understand what exactly it means on an implementation level,” Mr. Ferguson said.

Officials in New York and other states expressed similar fears and uncertainty. The office of Attorney General Letitia James of New York was aware of the updated guidance and reviewing it. Like in California, billions of dollars in grants could be on the chopping block.

Officials with the Federal Emergency Management Agency have already identified nearly $1 billion in funding they think should be cut and another almost $4 billion that is being reviewed, according to a memo reviewed by The New York Times.

Some funding — like $80 million from FEMA to New York City for migrant shelters — was clawed back this winter. The money for these beds came from FEMA’s Shelter and Services Program. Now federal officials want to cancel about $887 million in grants from that program.

Tricia McLaughlin, a spokeswoman for the Department of Homeland Security, said that the homeland security secretary, Kristi Noem, “has directed FEMA to implement additional controls to ensure that all grant money going out is consistent with law and does not go to fraud, waste or abuse.”

“The open borders gravy train is over, and there will not be a single penny spent that goes against the interest and safety of the American people,” Ms. McLaughlin added.

Another program under review involves nearly $2 billion in grants to law enforcement agencies, cities and states to help themselves prepare and respond to terrorist attacks. In the fiscal year ending in October, entities in New York received close to $320 million from this program.

Avi Small, the press secretary for Gov. Kathy Hochul of New York, pointed out in a statement that “the federal government has issued multiple orders, directives and guidance that were later overturned in court or rescinded by the administration.”

“We have received this guidance, which could have significant fiscal impacts for disaster relief and other critical state functions, and are assessing our options moving forward,” he added.

Last week, the New York State Education Department said it would not go along with the Trump administration’s threats to pull federal funding from public schools over certain diversity, equity and inclusion programs. The updated terms and conditions from D.H.S. for grantees includes new requirements in this area as well.

During Mr. Trump’s first term, his administration fought in court to withhold millions of dollars from law enforcement agencies in states and cities that did not cooperate with his immigration agenda. A federal judge ultimately overturned the executive order issued by Mr. Trump that withheld federal funding to sanctuary jurisdictions.

After wildfires erupted in California in 2018, Mr. Trump said on social media that he had ordered FEMA to “send no more money” unless the state changed its approach to forest management.

Mr. Trump has threatened on multiple occasions to withhold money from California, including earlier this year as parts of Los Angeles were ravaged by wildfires. He said that he would help but that he first wanted the state to impose voter identification laws and change its environmental policies.

On Friday, Trump officials were admonished by a federal judge for not complying with an order the judge had issued in early March to unfreeze billions in FEMA funds in at least 19 states. The Times reported last month that freezing of funds had caused chaos for state, local and nonprofits officials who were trying to respond to natural disasters.

Judge John J. McConnell Jr. of the Federal District Court in Rhode Island said on Friday that he thought the funding hold was a “covert” means to punish places with laws that prohibit cooperation with immigration officials.

“Trump has declared war against the American economy and our allies around the world,” said Jason Elliott, who previously served as Mr. Newsom’s deputy chief of staff. “Now American states, with millions of people who voted for him, are the target of his war against common sense.

“His funding threats against states, for nothing but political purposes, will directly harm millions of red voters in those purple and blue states.”

Laurel Rosenhall contributed reporting from Sacramento.