WASHINGTON (AP) — Immigration overhaul legislation has been dormant in the House for months, but a few Republicans are working behind the scenes to advance it at a time the Capitol is immersed in a partisan brawl over government spending and President Barack Obama’s health care law.

The chairman of the House Judiciary Committee, Rep. Bob Goodlatte, has been discussing possible legal status for the estimated 11 million immigrants living in the U.S. illegally. He’s also been working with House Majority Leader Eric Cantor, a fellow Virginia Republican, on a bill offering citizenship to immigrants brought illegally to the U.S. as children.

Reps. Raul Labrador, R-Idaho, and Ted Poe, R-Texas, are working on a plan to create a visa program allowing more lower-skilled workers into the country.

Goodlatte and the chairman of the House Homeland Security Committee, Rep. Mike McCaul, R-Texas, hold out hopes for floor action by late October on a series of immigration bills that already have passed their committees.

‘‘I would think that would be the next agenda item in the queue after we’re done with this mess,’’ McCaul said this past week, referring to bitter divisions over the health law, the level of government spending and the growing federal debt.

The attention of House GOP leaders seems certain to remain squarely focused on the fiscal disputes until they are resolved, leaving immigration on a back burner for some time to come. But lawmakers and outside advocates insist that three months after the Democratic-led Senate passed a sweeping immigration bill, the issue is showing signs of life in the Republican-run House.

‘‘Despite the appearance that would suggest everyone in Washington is focused on one thing, work is going on on other issues beneath the radar,’’ said Tamar Jacoby, head of ImmigrationWorks USA, a coalition of small businesses that supports comprehensive immigration legislation.

Goodlatte has made it clear he wants to see the issue solved.

Speaker John Boehner, R-Ohio, and other members of the House Republican leadership also support a resolution to an issue that has become a political drag for their party.

While Goodlatte has been outspoken about his desire to get legislation to the floor as soon as possible, House leaders have been more circumspect, adding to the uncertainty about whether or when anything actually will happen.

‘‘Moving immigration forward remains a priority, but right now there’s no firm timetable,’’ said Doug Heye, a spokesman for Cantor.

The issue is a top second-term goal for Obama.

House leaders have said they plan a step-by-step approach, in contrast to the comprehensive Senate bill that added billions of dollars in new spending on border security, remade the legal immigration system from top to bottom and created a 13-year path to citizenship for the millions living here illegally.

McCaul’s committee has approved a border security bill. Goodlatte’s committee has signed off on legislation addressing a range of issues, including visas for high-skilled workers and enforcement of immigration laws.

But so far, there’s been no House GOP bill taking on the trickiest policy issue for Republicans: what to do about those already here illegally.

A bill in the works by Cantor and Goodlatte would offer eventual citizenship to immigrants brought here as children. The proposal appears have support from a fair number of Republicans. But many Republicans are wary of backing anything broader that could be perceived as ‘‘amnesty’’ for people who broke U.S. immigration laws to be in this country.

There’s no guarantee House Republicans ever will offer a bill to resolve that issue, much less bring it to the floor for a vote.

Still, Goodlatte has outlined in some detail what he would like to see in such a bill, and his approach may contain the seeds of compromise.

Goodlatte would allow immigrants here illegally to obtain legal work status, and from there, they could use the existing routes to citizenship: marrying a U.S. citizen or getting sponsored by an employer or U.S. citizen relative. Such an approach would allow Republicans to deal with millions of people in the U.S. illegally without bestowing a so-called special path to citizenship as the Senate did — a concept that’s become toxic to many in the GOP.