Republican Rep. Raul Labrador on Thursday offered a potential immigration deal to the White House, saying the GOP would agree to loosen penalties on illegal immigrants if President Obama would agree to increase visas for foreigners who work in high-tech fields.

Mr. Labrador suggested dropping the penalty period that bars illegal immigrants from reapplying to enter the U.S. legally after being deported, a period that now lasts for between three and 10 years, depending on how long they had first remained in the country illegally.

“I think most Republicans agree that the 3- and 10-year bars have to go away because right now the people that are here illegally, they have to go home to become legal, but then they have to remain home for ten years,” the Idaho Republican said. “We remove those bars from them, you could fix the status of about 25 percent of the people that are here illegally right now if they return to their home country and then they come back legally.”

Under current law, those that leave the U.S. after living here illegally for six months or more cannot return for three years. Those who lived here illegally for at least a year cannot return for ten years. That has served as an incentive for many illegal immigrants to remain in the U.S. rather than return home and face yearslong bans.

Mr. Labrador, who dropped out of a bipartisan group last year that tried to strike a broad immigration deal, made the overture at a forum of House conservatives hosted by the Heritage Foundation.

He said in return for dropping the bars, Democrats should agree to boost legal immigration by granting green cards to foreigners who graduate from American universities with advanced degrees in the fields of science, engineering, math or technology.

Democrats have said they support that change, and indeed this week the Department of Homeland Security took unilateral steps to try to make it easier to approve visas for some high-skilled foreign workers.

But Mr. Labrador’s proposal didn’t impress immigration advocates, who said they will only accept a broad reform package that covers all parts of the immigration debate.

“This offer doesn’t come close to passing the laugh test,” said Frank Sharry, executive director of America’s Voice. “We want reform that includes legal status and the opportunity at citizenship for 11 million undocumented immigrants.”

“If House Republicans are prepared to give us a vote on that, they should give us a call,” Mr. Sharry said. “If not, they will feel the pressure of a changing electorate until they get on the right side of history.”

The advocates’ antipathy underscores the difficulty of moving any immigration legislation. Democrats want all parts of immigration tackled at the same time, and say a pathway to citizenship for most of the 11 million in the country illegally right now must be part of any deal.

House Republicans, though, say they won’t pass a massive bill, and say they want to tackle the immigration issue in pieces, including securing the border, boosting interior enforcement, rewriting the legal immigration system and expanding guest-worker programs.

Republicans also say they do not trust Mr. Obama enough to support any legislation that puts legalization before strict border security that he would have to enforce.

On issues ranging from immigration and drugs to Obamacare and gay marriage, Mr. Obama has unilaterally decided not to enforce or support numerous laws and provisions of laws with which he disagrees, a tactic Republicans have called lawless and a reason not to trust his “law-enforcement priorities.”

“I look back at the 28 years in the business world and I think, ‘Who was it that ever destroyed my trust that was ever able to re-earn it again?’ I can’t come up with a name in all that time,” Rep. Steve King, Iowa Republican and staunch illegal immigration opponent, said Thursday. “I don’t trust him, except to do what is politically to his advantage.”

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