The president’s forceful calls over the past week for bipartisanship on immigration failed to move the needle at all, as lawmakers began to look for a slimmed-down deal to protect Dreamers without broaching the big issues in the immigration debate.

One option floated by some Republicans on Capitol Hill is to offer Dreamers congressionally approved legal status, but not full citizenship rights, in exchange for limited border security improvements.

Although the proposal falls far short of what Dreamers have sought, Democrats wouldn’t have to accept big changes to legal immigration and conservatives wouldn’t have to vote for citizenship rights.

Sen. Marco Rubio, Florida Republican, said the slimmed-down version is becoming more likely even though he would like a bigger deal.

“We should have a plan B that basically focuses on border security and focuses on making sure that the people here under DACA now do not lose their status,” Mr. Rubio told CBS News.

Republicans have been discussing the plan in small group meetings on the Senate floor, saying the immigration deal has become a sliding scale. The more Democrats move toward security and limiting the chain of family migration, the more generous the legalization proposal can be.

The idea is likely to be a hot topic at the policy retreat in West Virginia, where House and Senate Republicans talk about their agenda for the year.

Democrats said President Trump’s State of the Union speech on Tuesday put a deal even further out of reach by showcasing black families whose teen daughters were slain by the illegal-immigrant-dominated MS-13 gang.

Democrats said they are still demanding full citizenship for as many illegal immigrants as possible and that is the one area of agreement on all sides.

“So why don’t we just do that: Build confidence, build bridges?” House Minority Leader Nancy Pelosi, California Democrat, said Wednesday.

But Sen. John Cornyn, the Texas Republican who is leading his party’s negotiations, said unless Democrats are willing to agree to a bill that goes beyond Dreamers to include security, there won’t be an agreement.

“That means the DACA program is going to fail. I don’t think anybody wants that to happen, so people need to quit taking hard and fast positions and realize that everybody is going to have to compromise to get to a solution,” he said.

Mr. Cornyn said Democrats need to move toward Mr. Trump.

“He’s the president. You’re not going to pass a bill or get it passed into law without his signature. So you can’t ignore him, [and] it’s going to have to be dealt with,” the senator said.

The president took his case directly to Democrats on Tuesday, laying out plans to link citizenship rights for 1.8 million Dreamers with action on his border wall, changes to chain migration and an end to the Diversity Visa Lottery. The proposal was met by Democrats with stony silence.

“For over 30 years, Washington has tried and failed to solve this problem. This Congress can be the one that finally makes it happen,” Mr. Trump said.

But Democrats bristled at his criticism of the MS-13 gang and accused him of trying to portray all immigrants as violent. They said that was a disservice to the dozens of Dreamers who had been guests of Democrats at the State of the Union speech.

After Mr. Trump finished his address, one young man, apparently a Dreamer, started heckling the president as he was leaving the chamber. The young man repeatedly called down from the gallery asking if he would sign a “clean Dream Act” — legislation that includes citizenship with no additional security.

House Democrats are pushing another bill, introduced with bipartisan support. It would couple a generous legalization of more than 3.2 million with calls for a study of border security, to be implemented over the next three years.

That bill, sponsored by Reps. Will Hurd, Texas Republican, and Pete Aguilar, California Democrat, would also send money to Central American countries to try to build their societies, hoping to stem another wave of illegal immigrants.

It does not include any money for a border wall.

“Sometimes, the simplest solution is best. Let’s not make it more complicated than it needs to be,” said Rep. Linda T. Sanchez, California Democrat. “Let’s pass that bill first. Let’s build some bipartisan trust and good will. Then we can have serious conversations about big infrastructure bills and comprehensive immigration reform.”

That bill is unlikely to meet with approval from Mr. Trump, who has said his border wall is a baseline for any proposal.

The president has even called a bipartisan Senate proposal that was much more robust than the Hurd-Aguilar plan too weak to sign.

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