House leaders plan to bring up a high-tech visa bill during the lame-duck session, teeing it up as the first test of the GOP’s post-election immigration strategy. An aide to Majority Leader Eric Cantor, R-Va., said the bill would come up “before the holidays.”

The bill (HR 6429) did not clear the House with the necessary two-thirds majority when it came up under suspension of the rules in September. But it has gained new life after an election in which Hispanic votes helped deliver President Barack Obama a second term, putting immigration in the spotlight.

The elections emboldened advocates and Democrats to call for a sweeping immigration overhaul that would grant citizenship to many of the nation’s 11 million illegal immigrants. Conservatives, though, were never going to support that approach. Speaker John A. Boehner, R-Ohio, sought to tamp down expectations on Nov. 9, saying he envisioned a “common-sense, step by step” approach to resolving the issue.

The first step is the revival of the high-tech visa legislation, which would grant permanent residency to foreign graduates of American universities with advanced degrees in science, technology, engineering and mathematics, known as the STEM fields.

The outcome could send a signal about the parties’ willingness to engage on the issue. If leaders manage to reach a bipartisan consensus, it could pave the way for more immigration agreements in the next Congress.

“This would be the first test,” said Tamar Jacoby, president and CEO of ImmigrationWorks USA, which advocates for immigration policy from a business perspective. “It’s no longer just about STEM. This is now going to raise all of the issues around, ‘Do we do [an immigration overhaul] all at once, or do we do this step-by-step?’ ”

Both parties are in rare agreement on this one facet of the immigration debate. In a Nov. 7 letter to colleagues, Cantor endorsed “reforming our immigration system to help American-educated entrepreneurs start and build businesses here rather than abroad.” And business groups have sought more green cards for high-tech workers for years.

“I think this is a critical start to building that consensus that Congress will need,” said Rebecca K. Peters, legislative affairs director at the American Council on International Personnel.

House Judiciary Chairman Lamar Smith, R-Texas, worked for more than a year to broker a deal on the STEM bill with Sen. Charles E. Schumer, D-N.Y., who chairs the Judiciary Subcommittee on Immigration, Refugees and Border Security. But the talks broke down over the summer, leading House leaders to bring Smith’s bill to the floor over objections of immigration advocates. Still, the bill got 30 Democratic votes.

Smith’s legislation would eliminate the diversity visa program, which awards visas to people from around the world via a lottery system, and redirect the 55,000 green cards it doles out every year to high-tech graduates. Democrats agreed to the swap, but they demanded several provisions to also help relatives of green card holders move to the United States.

It remains to be seen how far House Republicans will go during the lame duck to find common ground with Democrats on these family-related immigration initiatives.

One plan would allow the spouses and children of current holders of green cards to move to the United States without a work authorization until they get green cards of their own, according to lobbyists. Current rules force many families to live apart for several years while they await visas. Another proposal would split the 55,000 visas, with half going to high-tech foreign graduates and half to family reunification measures.

But even if the House is able to pass a bipartisan high-tech visa bill, there is no guarantee that it will lead to more immigration action. The all-consuming fiscal cliff fight is sure to cause bad feelings that could be hard to soothe.

House conservatives might also try to stop leadership efforts to reach out to Democrats. Boehner has already retreated from his Nov. 8 statement, in an ABC News interview, that “a comprehensive approach is long overdue” on immigration. The next day, he endorsed the more incremental approach.

Rep. John Fleming, R-La., chastised Boehner after his ABC interview for “getting ahead of House Republicans when he commits to getting a ‘comprehensive approach’ to immigration.”

Rep. Raúl R. Labrador, R-Idaho, a former immigration attorney who won his seat in 2010 with tea party backing, plans to introduce a series of narrow immigration bills that he says could get through the House.

“I don’t think there’s any room for a pathway to citizenship, but there is room for us to discuss a pathway for something like legal status,” he said.

Labrador warned, however, that Democrats shouldn’t overreach. “What I hope is that the Democrats don’t misread the election, and they think that what this means is we need a whole amnesty pathway-to-citizenship solution, without them sitting down with the Republicans who want to work on this issue,” he said.