Republicans ramped up their own demands in the reignited immigration debate, rolling out legislation Thursday that would cut levels of legal immigration and expand use of the E-Verify system to try to prevent illegal immigrants from taking jobs.

Rep. Lamar Smith, Texas Republican, said regardless of what Congress does on legal status for Dreamers, lawmakers must tackle the magnets that are spurring near-historic legals of immigration, both legal and illegal, into the U.S.

“These are bills that stand on their own and ought to be passed,” Mr. Smith said. “They prevent illegal workers from taking jobs meant for Americans and prioritize our legal immigration system to admit those who can best contribute to our economy.”

He said public polling has shown most Americans support an annual limit of 500,000 or less for legal permanent residents who get work visas, which one of the bills aims to accomplish by installing a points-based system that emphasizes English speaking and job skills for those who are admitted into the country.

The legislation would also limit family immigration to spouses and minor children of U.S. residents, and would create a new visa to allow citizens to care for their parents.

President Trump has already thrown his support behind the Senate version of the bill, known as the RAISE Act, which was introduced earlier this year by Republican Sens. Tom Cotton of Arkansas and David Perdue of Georgia.

The other piece of Mr. Smith’s proposal would expand E-Verify, the electronic system used to verify employees’ legal work status, for all new hires in the country. E-Verify is currently voluntary, but Mr. Trump’s 2018 budget blueprint included money to make it mandatory.

Mr. Smith said he’s confident he has the administration’s support, but wouldn’t forecast what the Senate would do if the bills clear the House, where Republicans have a more comfortable majority.

“I’m hopeful that they’ll do the right thing,” he said. “Every major provision in these two bills is supported by a majority of the American people, so I hope and expect the Senate to do what the American people [want] them to do.”

The GOP plans come as Democrats say they see an opening to pass a generous bill granting legal status to Dreamers in the wake of the Trump administration’s decision to revoke the legally questionable Deferred Action for Childhood Arrivals (DACA) program.

Mr. Cotton, sponsor of the Senate RAISE Act, said he’d be open to coupling his legislation with the Dream Act.

“I look forward to working with my colleagues to producing a legislative deal that the president can sign,” Mr. Cotton said.

But House Minority Leader Nancy Pelosi said Democrats won’t go for many add-ons, saying they see a consensus behind the legalization bill.

She said Mr. Trump assured her he would sign the Dream Act. Mrs. Pelosi said the president would probably want some sort of border security measures attached to the package but that any such deal would not include his desired U.S.-Mexico border wall.

Some Republicans on Capitol Hill have said that the border wall is the wrong fight, and the key to halting future illegal immigration is to limit the ability of unauthorized migrants to get jobs.

Rep. Steve King, Iowa Republican, said there is an appetite for such legislation, saying he’s managed to secure the attachment of several E-Verify-related amendments to an appropriations package the House is considering.

“That tells you a little bit about the commitment of this Congress to go forward with an E-Verify [that’s] mandatory,” Mr. King said.

He said he thinks there’s a good shot to get legislation out of the House and put it on the desk of Senate Majority Leader Mitch McConnell. “After that, it’s anybody’s guess,” Mr. King said.

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