Republican leaders have invited President Trump to visit Capitol Hill this week to try to repair the damage he did when he said he wouldn’t sign the immigration compromise that House Republicans had spent months writing.

Speaker Paul D. Ryan and his team want to have a vote on their immigration plan by the end of this week, but first they will need the president to tell them — and likely to tell the country as a whole — that he fully embraces the “moderate” bill that Republican leaders crafted, which combines a full pathway to citizenship for perhaps 1.8 million illegal immigrants with a border wall and changes to legal and illegal immigration policy.

On Friday, Mr. Trump said he wouldn’t sign it, undercutting weeks of intense negotiations in which his own administration had participated. The White House later insisted that Mr. Trump misunderstood the question and would sign the bill, which was written with his own aides’ involvement and crafted specifically to match his four pillars of immigration reform.

But Mr. Trump has yet to recant his rejection personally, leaving conservatives wary of being left out on a limb if they back an “amnesty” without having Mr. Trump to provide political cover with their right-wing base.

Indeed, they are getting pressure from the right. Former White House political strategist Steve Bannon warned of impending political catastrophe for Republicans if they advance the moderate bill.

“This is a way [to] potentially lose 50 seats,” Mr. Bannon told ABC’s “This Week” program Sunday.

Republicans won’t get any help from Democrats, who appear unified in opposition. They say they will reject funding for a border wall that they call effectively useless and symbolically counterproductive and criticize Mr. Trump’s policies as increasingly cruel.

Democrats fanned out across the country on Sunday, which was Father’s Day, to denounce the Trump administration’s rate of deportations — still only about half of what it was during the peak years under President Obama.

The latest target is the administration’s zero-tolerance policy, which is applying criminal penalties to illegal immigrants nabbed at the border. When parents who come with children are arrested for jumping the border, it means at least a short-term separation while the parent goes to jail for a few days.

The issue has become convoluted, with both sides delivering misleading versions of what is a complex system of polices and laws governing how the country treats illegal immigrants who arrive as adults, as families, and as juveniles traveling without their parents.

On Sunday, Democrats, with camera crews in tow, visited a U.S. Immigration and Customs Enforcement detention facility in New Jersey and a Customs and Border Protection facility on the border in Texas. In each case, they said they were there to protest the zero-tolerance policy.

“What we saw here today was heartbreaking,” Rep. Bill Pascrell Jr. of New Jersey said after he emerged from the ICE facility in his state. “Parents are being held prisoner inside, literally crying to us to be reunited with their families.”

He called immigration detention “a sin.”

The issue deeply divides Republicans, who are increasingly breaking with the administration over the zero-tolerance policy.

The administration has said if people looking to claim asylum show up at an official port of entry with their children, then they won’t be separated. That policy applies to thousands of cases each month.

But two Republicans on Sunday said they knew of two cases in which a parent who attempted to show up at an official entry point — just as the administration recommends — and still had their children separated.

Sens. Jeff Flake of Arizona and Susan M. Collins of Maine sent a letter challenging the truthfulness of the administration and demanding answers.

“What we do know is this: The secretary of homeland security testified that if parents present at a legal port of entry with their children with a claim of asylum, that the children would not be taken away,” Ms. Collins said on CBS’s “Face the Nation.” “Yet there are numerous credible media accounts showing that exactly that is happening. And the administration needs to put an end to that right off.”

Ms. Collins said she doubted the Trump administration policy would be successful in curtailing illegal immigration.

Given the relatively recent nature of the zero-tolerance policy and time it takes illegal immigrants from Central America to make their way north, the administration should start to see results — or evidence of failure — over the next several weeks.

That won’t be in time to inform this week’s House debate.

Republican leaders plan to put two bills on the floor. One, sponsored by Rep. Bob Goodlatte of Virginia, would write the Obama-era DACA program into law, giving it more permanence, and would include a massive burst of enforcement measures, mandatory use of E-Verify, more Border Patrol agents and making overstaying a visa a crime.

The other, the Republican leaders’ plan, would combine a full pathway to citizenship for DACA recipients and perhaps 1 million other illegal immigrants with border wall money, limits to chain migration, an end to the visa lottery and some measures to speed up deportations.

Mr. Trump, in an interview with Fox News on Friday, threw cold water on that plan, which he called the “moderate” one.

“I am looking at both of them,” Mr. Trump said on “Fox and Friends.” “I certainly wouldn’t sign the more moderate one. I need a bill that gives this country tremendous border security. I have to have that.”

For hours, officials ignored requests to clarify Mr. Trump’s remarks.

On Friday evening, eight hours after Mr. Trump’s comments, the White House seemed to settle on a position, saying Mr. Trump misunderstood the question and would sign the Republican leaders’ “moderate” bill.

“The president fully supports both the Goodlatte bill and the House leadership bill,” said Raj Shah, principal deputy press secretary at the White House.

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