House Republicans may be back home on vacation this week, but they are still facing pressure to join the rebellion against their party leaders and sign onto the petition drive to have Congress vote on granting citizenship rights to illegal immigrant “Dreamers.”

Some of the party’s biggest financial backers are trying to coax more Republicans to buck their leaders and sign the petition, as are editorial boards and immigrant rights groups in home districts.

The dozens of Republicans targeted by the pressure campaign, though, have remained mostly silent. They are hoping to stay out of a fight in which many of them have to pick between President Trump on one side or major Republican Party donors like the Koch brothers on the other.

“You have to show that you are connecting with your base, you are listening with your base, but you also don’t want to get into a confrontation with the president,” said Alfonso Aguilar, president of the Latino Partnership for Conservative Principles and an opponent of the petition effort.

The petition drive is being led by a group of Republicans determined to force a vote on the fate of Dreamers. The cabal of lawmakers, each facing competitive re-election races, is led by Reps. Will Hurd of Texas, Carlos Curbelo of Florida and Jeff Denham and David G. Valadao, both of California.

Their petition, if successful, would force Republican leaders to bring to the floor an immigration bill and then hold votes on four competing proposals. The bill that gets the most votes would be the winner.

The likely victor would be a plan sponsored by Mr. Hurd that would grant a generous pathway to citizenship to millions of Dreamers in exchange for a future study of border security needs. Other options would include an enforcement-first plan that would fund the president’s border wall.

Backers have amassed 213 signatures, leaving them five short of the threshold for success.

Of those 213, 23 are Republicans. The Republican rebels say they should be able to collect many more easily once Congress returns from its 10-day Memorial Day vacation.

“Regardless of political affiliation, everyone should support a transparent debate that puts all our options on the table for consideration,” Mr. Hurd said. “I remain committed to securing 218 signatures and debating the issue in front of the American people.”

All but three Democrats have signed onto the effort.

The holdouts, Reps. Henry Cuellar, Filemon Vela and Vicente Gonzales, are all from Texas. They say they fear the process could yield a deal that does more harm than good for border communities if it leads to a border wall.

“Representatives Vela, Gonzalez, and I are saying ‘no’ to signing the petition,” Mr. Cuellar said in a statement. “As I’ve said in the past, I cannot support a border wall which is a 14th-century solution to a 21st-century problem. There are better ways at finding solutions … and I will continue to work in a bipartisan way in Congress to properly address these issues.”

Their opposition has helped turn up the heat on about a dozen moderate Republicans holdouts who have supported like-minded efforts in the past.

Business groups have taken the lead in pushing the Republican Party.

“The U.S. Chamber has been actively engaged with members on efforts to start a much-needed debate on immigration issues in the House of Representatives,” the chamber said in a statement to The Washington Times. “We have issued statements of support to those members who have signed the discharge petition and have made clear that we will work with interested members on any viable path to advance bipartisan legislation that protects Dreamers from the threat of deportation and institutes much-needed improvements to our nation’s border security efforts.”

FWD.us, an advocacy group created by Facebook’s Mark Zuckerberg to push for an immigration overhaul, is urging voters to call on their House representatives to join the push.

David MacNeil, founder of WeatherTech and a major donor to Mr. Trump’s presidential campaign, told Politico last week that he will give no more money to lawmakers who don’t back legal status for Dreamers.

Many of the Republicans being targeted have remained mum, with their offices not responding to inquiries this week.

Other targeted Republicans have said they won’t sign on because they want to give their party’s leaders a chance to find a compromise.

Jacinta Ma, director of policy and advocacy for the National Immigration Forum, which is working to line up more petition signatures, said these Republican lawmakers have various reasons.

“Some people, I think, are more traditional and they want it to go through committee, be decided by committee and then have an opportunity to vote for it on the floor when it goes through the whole process,” Ms. Ma said.

“They may need to be in the good graces of Republican leaders,” she said. “Maybe they have a close election coming up and they need those leaders to fundraise for them or endorse them. Maybe someone wants to be part of leadership.”

Yet Republican leaders have been easy on members who have signed the petition. The party continues to send resources their way, and there has been no hint that they will lose committee posts or other leadership positions.

Mr. Aguilar said he is concerned that Republicans supporting the discharge petition are playing right into the hands of Democrats. He agreed with House Speaker Paul D. Ryan, Wisconsin Republican, who has said any bill that emerges from the discharge petition process is likely to draw a veto from Mr. Trump.

“Without the White House involved, I don’t see how you can get anything done,” said Mr. Aguilar, adding that Democrats are laying an election trap for Republicans. “At this point, the angle of Democrats is to pass something so they can blame Republicans. So why do we want to be part of that?”

Trying to head off the petition drive, Mr. Ryan and his fellow leaders have said they will bring unspecified legislation to the floor by June 22.

That was good enough for Rep. Joe Barton, Texas Republican, one of the targets for signing the petition.

Still, he said, the petition drive has already worked.

“The fact that a significant number of the majority party were willing to sign the discharge petition has forced the leadership, with strong encouragement from the Freedom Caucus … to address the issue.”

If the petition drive succeeds, the earliest it could force a floor debate would be near the end of June.

That gives Republican leaders several weeks to try to write their alternative that can get majority support.

But there are major questions as to whether the leaders can bridge a gap between conservatives who say any legal status for Dreamers is an amnesty and more moderate Republicans who want general citizenship rights for millions of illegal immigrants with only minimal security.

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