Democrats blocked an attempt Tuesday to crack down on sanctuary cities as they tried to steer the Senate’s immigration debate toward their goal of legalizing “Dreamers” and away from major new security or policy changes that might upend the current immigration system.

After weeks of complaining about the lack of time to debate immigration, senators are now two days deep into the debate and have yet to raise a single proposal on the chamber floor.

Republican leaders said the lack of action showed Democrats were looking to score political points, but Democrats said they’re waiting for a bipartisan deal to emerge, and don’t want the Senate to stray beyond strict boundaries of amnesty for Dreamers and limited border security enhancements.

“We want to do two things: protect Dreamers and get 60 votes,” said Senate Minority Leader Charles E. Schumer, New York Democrat. “It’s not an easy needle to thread. But we’re making good headway.”

Majority Leader Mitch McConnell said he’s giving senators this week only to debate immigration, saying they’ve been talking about the matter for months and should have been itching to go with their ideas.

“We’ll need to wrap this up this week,” the Kentucky Republican said. “If there are 60 votes for any of these proposals here in the Senate we should be able to discover that.”

Republicans said they were surprised Democrats were blocking attempts to vote on amendments Tuesday, after clamoring for this debate.

“We’re going to find out real soon whether or not the Democrats are serious — whether or not they want a solution or whether they just want a campaign issue,” said Sen. John Thune, South Dakota Republican.

Part of the problem was that Democrats had no clear proposal to rally around.

Instead, they’re counting on bipartisan negotiations being led by Sens. Susan Collins, a Republican, and Joe Manchin III, a Democrat, to produce a new “grand bargain” deal that could cut through the fractious nature of the immigration debate.

Those talks have yet to conclude, but negotiators were looking at a deal that would include legal status for illegal immigrant “Dreamers” with piles of cash for border security.

Such grand bargains have been a staple of past immigration debates, but none of them have ever become law. A grand bargain would also short-circuit the freewheeling floor fight all sides had promised.

Republicans, for their part, can agree on border security changes. But they’ve struggled for consensus on how big the amnesty for Dreamers should be, and on whether to also include cuts to chain migration, as Mr. Trump has demanded.

As of Tuesday afternoon, Republicans had filed more than a dozen amendments ranging from sanctuary cities and making English the official language of the U.S. to criminalizing those who overstay their visas.

Democrats, by contrast, had offered just one bipartisan bill with a generous amnesty for Dreamers, coupled with a promise of border security by 2021.

Mr. Schumer suggested bringing both that plan, sponsored by Sen. Chris Coons, and a GOP plan enshrining Mr. Trump’s four-pillar framework to the floor for votes. He said he doubted either could clear the 60-vote threshold the Senate has set for action, but he said at least then the parameters of the debate would be clear.

Sen. Bill Nelson, Florida Democrat, was skeptical about finishing an immigration bill by the end of the week but said senators are known for striking deals in time for a recess week. The Senate is slated to be on vacation next week to commemorate the George Washington’s birthday holiday.

“I think come Thursday afternoon, they are gong to start smelling the jet fumes,” Mr. Nelson said.

He said Democrats will likely unify around a proposal originally offered by Sens. Richard Durbin, a Democrat, and Lindsey Graham, a Republican, that would couple an amnesty with a small downpayment on the border wall and changes to the Diversity Visa Lottery.

But Mr. Graham wouldn’t commit to even offering that plan as an alternative. He said he was working on a GOP bill that enshrines Mr. Trump’s framework, including citizenship rights for up to 1.8 million Dreamers, a $25 billion border wall trust fund, an end to the Diversity Visa Lottery and major limits on the chain of family migration.

Mr. Graham called that “a serious offer,” though he said the cuts to family migration have to be made up with increases elsewhere, so overall legal immigration doesn’t decline.

“The way you pass a bill like this is you take off the table what won’t work and what’s left is what you probably can get 60 votes for,” said Mr. Graham. “So we are begging to refine whether or not it is a two-pillar bill or a four-pillar bill and what are the combinations that would allow it to be four-pillar,” he said.

Sen. Jeff Flake, who had been working with Mr. Graham and Mr. Durbin on legalization proposals earlier, floated his own plan Tuesday that tracks Mr. Trump’s framework. It would couple a more generous amnesty with border fencing and limits to family migration. Instead, it would boost skills- and education-based immigration.

“One of the biggest things is it doesn’t restrict legal immigration. It keeps current levels. That’s it,” said Mr. Flake. “Cutting legal immigration in half or by a third is just a nonstarter for me and a lot of Republicans and certainly all the Democrats.”

The U.S. Chamber of Commerce weighed in on Tuesday, calling for an even broader solution that not only legalized Dreamers but also created permanent legal status for hundreds of thousands of would-be illegal immigrants here under temporary humanitarian protections.

The chamber did say it would welcome “responsible border security efforts” but said any move to require businesses to check their workers through E-Verify cannot be adopted until there’s also a new guest-worker program to staff farms.

And chamber officials said they will resist any effort to cut the overall level of legal immigration, signaling that any green cards cut from family-based migration should be reallocated to business needs.

“A reduction in legal immigration will hinder overall economic growth and only encourage additional illegal immigration,” the chamber said.

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