This approach has benefits for Mr. Trump and his party. The president keeps a key campaign promise by not agreeing to anything his base would consider too lenient.
He avoids an assault from the conservative news media, which called him names like “Amnesty Don” after he showed signs of flexibility.
Senator Lindsey Graham of South Carolina, a Republican who has been a frequent target of the right for trying to broker bipartisan agreements on immigration, said the current debate was noticeably lacking in the bitterness that had so easily whipped up against Republicans in the past.
“That’s just on the margins now,” he said. “Trump introduced a 1.8 million-person pathway to citizenship. And it’s hard to get to his right on this stuff.”
And if the efforts in the Senate to pass a deal go nowhere, that saves vulnerable incumbent Republicans in the House from having to take a difficult vote and risk a primary challenge from the right. Before the end of March, 25 states have filing deadlines for House and Senate races, leaving ample time for conservative challengers to emerge.
“There are no profiles in courage during filing season,” said Richard F. Hohlt, a veteran Republican consultant. “If you’re up there — in the House especially — you’re saying, ‘Why do I want to get in the middle of this debate if I don’t have to?’”
“It’s all about the self-interest of the members,” he added.
In the 32 years since the nation’s immigration laws were last overhauled, efforts to take another run at it have followed a predictable dead-end path. Both times the Senate voted on immigration legislation since — in 2006, at President George W. Bush’s urging, and seven years later, when Barack Obama was in the White House — the majority of Republicans in the chamber voted no.
Credit Joshua Roberts/Reuters
Senator James Lankford, a conservative Republican from Oklahoma who has been trying to bring together the divergent factions in his party and resolve the status of the young immigrants, known as Dreamers, said he found the repeated failure to address such a glaring problem incredibly frustrating.
“It’s not that it hasn’t been something obvious to everyone,” he said. “It’s that Congress has been unwilling to address it.”
“For 20 years, there have been groups that have pushed us apart, saying don’t talk,” he added.
There are several factors that make proponents of a compromise hopeful that this time is different, though they acknowledge the odds are against that. First is the pressure of a deadline. Mr. Trump has said that he will let legal protections for the Dreamers expire on March 5, and the Senate has a self-imposed end-of-the-week deadline to write legislation.
With the exception of a small group of hard-liners on the right, conservatives agree that Congress should do something to save the Dreamers from living with the constant fear of deportation.
And absent an agreement now, it is difficult to see how Mr. Trump would ultimately achieve any of the demands in his plan, like billions in funding for his border wall and an end to the visa program known as the diversity lottery, which allows in immigrants from countries that do not send many people to the United States. Mr. Trump has called it a glaring loophole for potentially dangerous immigrants.
“These conservatives who say they’re so worried about terrorism, this should be a no-brainer,” said Travis Korson, a Republican consultant who is working to push more Republicans to agree on a Dreamer solution.
Evangelical groups are now pressing lawmakers to reach a solution, giving the issue the imprimatur of one of the most important constituencies in the conservative coalition.
In a letter to Mr. Trump and members of Congress last week, dozens of church leaders called for action. “The Bible speaks clearly and repeatedly to God’s love and concern for the vulnerable,” they wrote, “and also challenges us to think beyond our nationality, ethnicity or religion when loving our neighbor.”
Alex Conant, a Republican strategist who was an adviser to Senator Marco Rubio of Florida when Mr. Rubio was pilloried by the right for helping write a comprehensive immigration overhaul in 2013, noted that with Republicans in control of the White House and both houses of Congress, there may not be a better opportunity for years.
Mr. Trump, he said, “could become the first president since Reagan to fix immigration.”
Credit Tom Brenner/The New York Times
“He has the chance to sign something historic,” Mr. Conant added. “And if I were advising him that’s what I’d tell him to do. But I’m not advising him. Stephen Miller is.” (Mr. Miller is the senior policy adviser who, perhaps more than anyone else at the White House, has been a check on the president’s impulses to consider a deal that is less restrictive.)
In many ways, the Republican Party is still stuck in the same debate it has been having for a generation over law and order versus “amnesty,” a generic term many on the right have used to describe plans that would provide legal status or citizenship to anyone here illegally.
It was not always such a loaded term. Speaking in 1984, President Ronald Reagan described his views this way: “I believe in the idea of amnesty for those who have put down roots and who have lived here, even though sometime back they may have entered illegally.”
But with the election of a president who ran on the most restrictive immigration platform in modern times, the reality for the Republican Party under Mr. Trump is more difficult than it was under Reagan, who still elicits disapproval from conservatives who believe he gave too much away in the immigration overhaul of 1986.
“We have gone through this before with President Reagan, and we have been suffering the consequences since,” said Jenny Beth Martin, a co-founder of the Tea Party Patriots.
“Prove to us that you are worthy of our trust,” she added, “and then come back and talk to us about how we deal with people who are entering this country illegally.”