Anger at the Obama administration’s lax immigration policies boiled over Thursday as Democrats and Republicans demanded that the State Department punish countries that refuse to take back their citizens living in the U.S. illegally — often letting dangerous criminals run free on American streets.

Republicans demanded that the State Department make an example of some of the 23 most problematic countries by stripping their visas, as U.S. law allows.

Democrats put further pressure on the Obama administration, saying Congress should cancel foreign aid to any country that regularly refuses to cooperate with U.S. deportation.

“I think they will, in a heartbeat, they will respond to that,” said Rep. Stephen F. Lynch, Massachusetts Democrat.

The issue of criminal immigrants on the streets unified lawmakers, who are usually riven over the immigration issue.

A case last year in Connecticut put a terrifying point on the matter. Haiti refused to accept one of its citizens, Jean Jacques, who had served time in the U.S. for attempted murder. Immigration authorities were unable to deport Jacques, and the U.S. released him. Months later, Jacques killed a young woman in an apparent dispute with her boyfriend over drugs.

Daniel Ragsdale, deputy director of U.S. Immigration and Customs Enforcement, which oversees deportations, said authorities want to deport tens of thousands of criminals whose home countries won’t take them back.

Some 35,000 alone are from Cuba, and 1,900 are from China, according to ICE figures.

Under a section of U.S. immigration law, officials can retaliate by withholding visas from countries deemed to be recalcitrant.

That penalty was used once, in 2001, against Guyana, which had built a backlog of 113 people it refused to take back. Within two months after the U.S. stopped accepting visas, Guyana yielded and cleared 112 for travel.

Despite that stunning success rate, the Obama administration has not used the tool.

The State Department blamed the Department of Homeland Security, saying Secretary Jeh Johnson needs to officially request that visas be withheld. If Mr. Johnson issues notification, the State Department is required under the law to halt visas, Michele Bond, assistant secretary for consular affairs, told the House Oversight and Government Reform Committee.

Mr. Ragsdale acknowledged that Mr. Johnson had not made any requests.

An inspector general’s report last month said ICE officials don’t think the State Department would take the issue seriously. Ms. Bond told the committee Thursday that her department doesn’t have a choice if Mr. Johnson acts.

His top aides have threatened to issue the notification, but Mr. Johnson hasn’t taken the key step.

“It has not been imposed, but it has been discussed and countries know they are on the line,” Ms. Bond said.

She said the threat alone of visa penalties has forced better cooperation from some countries but too much pressure might cause retaliation with visa denials to U.S. travelers.

“The likely response of any government — one response — would be that they would reciprocate,” she said.

Democrats said cases like Jacques’ are tragic and ruin chances to strike broader immigration deals in Congress because they reinforce the notion that the U.S. doesn’t have control of its immigration system.

With criminals released, Republicans said, the administration has American blood on its hands.

“That’s on you. They shouldn’t be here in the United States of America,” committee Chairman Jason Chaffetz, Utah Republican, told Ms. Bond and Mr. Ragsdale. “You’re so worried about playing nice instead of implementing the law, these people are committing more crimes. I just got through listing the people — everything from murder to DUI to sexual abuse. Get rid of them!”

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