Advocates of an immigration overhaul may see a silver lining in the recent 16-day government shutdown for their cause in the House, sensing that Republicans will want to win back some of the popular support they lost over the past month.

But House GOP insiders remain skeptical that the fractured Republican Conference will be able to get something done on the issue anytime soon.

Backers of a rewrite of immigration laws view the recent plunge in Republican favorability ratings as an opening for the party to push the immigration issue and help repair the GOP brand. Even so, action may be stymied by continued infighting over government spending and lawmakers who have little interest in helping the party resuscitate its damaged reputation.

Sources familiar with the thinking of Republican leaders and the rowdy conservatives who clash with them are pessimistic about the party’s ability to come together, even if immigration changes enjoy popular support with voters and among the party.

“I would ask these immigration proponents, ‘Does our party look like it’s doing a good job of actively managing our favorables?’” one GOP aide questioned. “Does the [Republican National Committee] want us to do something? Sure it does, to give them talking points headed into 2016, making the party look like it’s more reasonable and in tune with demographics a Republican presidential candidate might need. Is that something that’s actually viable in the House? No. It’s not.”

Another Republican aide predicted, “There is no chance the House brings anything to a vote. I’m pretty confident you don’t have anyone in Republican leadership in the House telling you it would be good to vote on it. Just not going to happen, no matter how much [the president] wants to change the debate to that issue.”

Then there’s the timing of the next budget fight.

The bipartisan deal to reopen the government — requiring a budget conference report in December, a continuing resolution by mid-January and an increase in the debt limit in February — may prevent the House from acting on any immigration bills until spring 2014 at the earliest, sources on both sides say.

While Republicans may have reason to worry about the 2014 midterms, immigration supporters say passage of an overhaul is more important for the GOP’s ability to take back the White House in the 2016 election. They contend that extends the window for House action into next year.

“It’s almost impossible to understate how important this is for 2016; [Republicans] can afford to slip no further,” said an operative working on immigration issues. “When you look at the national party, there’s no one who wants to elect a Republican president who doesn’t think they should do this. It’s not that they’re going to start winning tons of [Hispanic] votes, it’s that they’re going to lose less of them and they’re going to be able to compete.”

This operative noted that party leaders are concerned about the political shifts in the Southwest — where swing states such as New Mexico and Colorado have become increasingly Democratic — and noted that it would only take four or five key GOP lawmakers to impress on their colleagues that reality.

House Judiciary Chairman Robert W. Goodlatte, R-Va., and former vice presidential nominee Paul D. Ryan, R-Wis., could be important to that effort.

Several piecemeal immigration bills have already been approved by House committees, and the best-case scenario for immigration overhaul backers may be to pass as many as possible and then get into a formal (or informal) conference with the Senate on its bipartisan bill. The thinking goes: If they can get to a conference, they may be able to approve something that implements changes to the system, even if it’s not as comprehensive as the Senate measure.

Several immigration supporters also say the shutdown marginalized many of the GOP’s most conservative voices, exposing a rift that otherwise could have been outed in an immigration debate.

In the Senate, backers are resigned to hoping that Speaker John A. Boehner, R-Ohio, and Majority Leader Eric Cantor, R-Va., will bring legislation to the floor.

This weekend, Senate Majority Leader Harry Reid, D-Nev., gave a lukewarm response to a question about an immigration bill’s chances. He said Boehner should break away from the “Hastert rule” requiring a majority of Republican support for any bill — just as he did with the measure that reopened the government and extended the debt ceiling.

“His history was, what I’ll do is, I’m not going to pass anything unless it’s the majority of the majority. I hope this breaks that,” Reid told the Spanish-language network Univision. “If immigration were brought to the floor tomorrow it would pass … in the House of Representatives, overwhelmingly. The American people want it. It would reduce the debt by a trillion dollars. It’s long overdue.”

Reid added that it’s a “two-year Congress” and that he won’t support any bill without an explicit pathway to citizenship, even though advocates acknowledge they have a better chance at getting something into law if they pass other fixes to the system.

Outside groups are going to continue to actively push for legislation, hoping they can influence what those inside the Capitol believe is an unmovable House Republican Conference. U.S. Chamber of Commerce President Tom Donohue said Monday at a Christian Science Monitor breakfast that he is still confident a bill would get done this Congress, and a group of 300 advocates from 50 congressional districts will descend on Washington next week — sponsored principally by Donohue’s group, Bibles, Badges and Businesses for Immigration Reform, FWD.us, the Partnership for a New American Economy — to lobby House Republicans for changes.

“The president, after the [shutdown] deal was inked, his first comments were that he wanted to move on with a number of issues and the first thing he said was an immigration bill,” Donohue said.

Donohue suggested another way the current budget fights could clear a path for action on immigration.

“He will also advance it in part by getting involved and helping us to come to a satisfactory and progressive — meaning moving forward — set of solutions on tax and spending and on entitlements,” he said.