House Republicans on Wednesday accused the Democratic mayors of New York, Denver, Boston and Chicago of harboring criminal immigrants in an acrimonious congressional hearing over what role large cities should have in the Trump administration’s immigration crackdown.

Under fiery and angry questioning from Republicans on the House Oversight Committee, the mayors defended their policies and their cities’ efforts to house and feed migrants, tens of thousands of whom were bused to their communities by Republican governors. The mayors rejected the notion that the local police should help in the administration’s deportation efforts.

“We do not have the capacity for our law enforcement to be doing federal immigration enforcement,” Mayor Mike Johnston of Denver told lawmakers. “But we want to be partners in making sure we are pulling violent criminals off the street.”

Mr. Johnston spoke of the influx of 42,000 migrants two years ago, many bused from Texas, “mostly women and children in 10-degree weather with only sandals and a T-shirt.”

The hearings seemed to capture the political moment. It was a clash of the law-and-order Republican Party led by President Trump and liberal politicians running cities, broadly known as sanctuary cities, that have large populations of immigrants.

Even the language employed by both sides underlined the stark differences in the ways the two parties approach the issue. The chairman of the committee spoke of “illegal aliens,” a term now out of favor among Democratic leaders and immigrant advocates, who prefer the term “undocumented.”

At the heart of the hearing was a question seemingly unique to America’s decentralized political system — the extent to which one segment of government is allowed to curb its cooperation with another.

Republicans are seeking a more proactive approach by cities, saying that local police departments should be doing more to facilitate the transfer of undocumented immigrants to the federal authorities. Democrats counter that if cities were deputized to help enforce federal immigration laws, they would have to divert resources away from other priorities, such as investigating crimes and apprehending violent criminals.

“There is no place in America, not one, that actually provides sanctuary from federal law,” Representative Dave Min of California, a Democrat, said at the hearing. “The real issue here is whether state and local governments should spend scarce taxpayer dollars to help the federal government enforce its immigration priorities.”

Republicans repeatedly sought to induce the mayors into the kind of stumbles that derailed the presidents of Harvard and the University of Pennsylvania, who lost their jobs after testifying before Congress about campus antisemitism.

At the start of the hearings, the committee’s chairman, Representative James Comer, Republican of Kentucky, referred to a number of cases where undocumented immigrants were charged with rape or murder. He asked the mayors whether they would hand over the “criminal” to the federal Immigration and Customs Enforcement, or ICE.

“Our local law enforcement works hard every day to get criminals off the streets of Chicago,” replied Mayor Brandon Johnson of Chicago.

Mr. Comer interjected, “Will you turn that criminal over to ICE?”

“We do not harbor criminals,” Mr. Johnson said, adding details of criminal procedure without directly answering the question.

The exchange ended with Mr. Comer concluding that it proved that Democratic mayors were shielding criminals from federal law enforcement.

“We have to have cooperation,” he said.

The past several years have seen record levels of unauthorized immigration into the United States, and Americans have become more supportive of stemming the flow. A poll from The New York Times and Ipsos in January found that a vast majority of Americans — 87 percent — supported deporting undocumented immigrants with a criminal record. A majority said they favored “deporting all immigrants who are here illegally.”

Republicans stepped up their confrontational tone as the hearings wore on, with Representative Nancy Mace, Republican of South Carolina, asking questions like whether the mayors hated President Trump more than they loved their country. She accused the mayors of having “blood on your hands.”

A number of Republican lawmakers who spoke at the hearing said the mayors were violating federal laws and should be prosecuted. They cited specific cases where people had been attacked or killed by undocumented migrants, suggesting cities’ immigration polices had contributed to those crimes.

“Every one of you is exposed to criminal culpability here,” Andy Biggs, Republican of Arizona, told the mayors. “That’s the reality of it.”

Several times in the hearings the mayors turned Republicans’ accusations back on them to try to score their own political points, urging them to focus on the economy, pass immigration reform, strengthen gun laws and oppose other Trump administration policies the mayors described as harmful.

“If you wanted to make us safe, pass gun reforms,” Mayor Michelle Wu of Boston said.

At another point, Ms. Wu spoke to the damage that deporting large numbers of migrants would have on cities.

“I do not support mass deportation,” she said. “That would be devastating for our economy. There are millions of people who are running our small businesses, going to our schools.”

Mayor Eric Adams of New York, who has talked of allowing ICE into jails in spite of his city’s sanctuary laws, did not come under the same level of scrutiny from Republicans.

But he was questioned by Democrats about accusations that he had engaged in a quid pro quo with the Trump administration, which dropped federal corruption charges against him, saying that it needed his help on immigration. Representative Robert Garcia, Democrat of California, called on Mr. Adams to resign; the mayor said he had done nothing wrong.

Mr. Adams grew weary of the line of questioning. “It appears as though we’re asking the same questions over and over and over again,” he told Representative Jasmine Crockett, Democrat of Texas. “My comments are not going to change. No quid pro quo, no agreement. I did nothing wrong.”

Although Republicans spared Mr. Adams from the most aggressive questioning, the vision he laid out did not fully jibe with the Trump administration’s talking points.

“I must create an atmosphere that allows every law-abiding resident, documented or not, to access vital services without fear of being turned over to federal authorities,” he said during the hearing.

Emma G. Fitzsimmons and Maya C. Miller reported from Washington and Thomas Fuller from San Francisco. Jack Healy and Patricia Mazzei contributed reporting.