While House GOP leaders insist the Senate version is dead on arrival, some signal possible areas of compromise on the measure considered vital to both parties and the legacy of President Barack Obama.

For Obama and Democrats, passage means following through on a promise to the nation’s largest minority demographic — Hispanic Americans — to reduce uncertainty for millions of them living in the United States illegally.

For Republicans, it is the best and perhaps lone chance to make inroads with the key Hispanic demographic that overwhelmingly supported Obama in last year’s presidential election.

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However, Republicans in Congress and across the country remain split over how to deal with the 11 million immigrants living illegally in America — those who sneaked in or overstayed visas.

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That issue will be significant in next year’s congressional elections with House Republicans under pressure from conservatives to oppose what the political right-wing calls amnesty in the form of a path to legal status and even citizenship for undocumented immigrants.

Moderate Republicans, including most of the party’s leaders in Congress, want a compromise that would represent a potential breakthrough for the GOP with Hispanic Americans.

Former President George W. Bush added his voice on Wednesday to calls for a solution, telling a ceremony for new citizens that the present system was broken.

“I do hope there is a positive resolution to the debate,” Bush said in a rare public comment about politics since he left office. While he refrained from taking sides, he sounded moderate in saying he hoped that “we keep a benevolent spirit in mind.”

“We understand the contributions immigrants make to our country,” said Bush, a Republican. “We must remember that the vast majority of immigrants are decent people who work hard and support their families and practice their faith and lead responsible lives.”

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The 1,000-plus page Senate immigration bill passed last month tales a comprehensive approach that removes the threat of immediate deportation for most undocumented immigrants and provides a path to legal status and eventual citizenship.

It also includes stronger enforcement provisions including the virtual militarization of the Mexican border with almost double the agents, new technology and a longer fence, as well as requiring employers to verify the immigration status of workers and improved monitoring of visa holders.

While the Senate measure passed 68-32 with support from 14 Republicans, House GOP leaders insist they will not bring it up because most of their caucus opposes it.

House Speaker John Boehner has repeatedly said any immigration bill brought up for a House vote must have support from a majority of Republicans.

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Democrats, meanwhile, call on Boehner to allow a House vote on the Senate version, believing it could pass with unified Democratic support along with several dozen Republicans.

Rep. Steny Hoyer of Maryland, the No. 2 Democrat in the House, said Wednesday that Republicans were “splintered and confused” on immigration, describing what he called a “deeply divided party” that was “much more into ideology than they are into solving problems.”

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The White House sought to keep up pressure on House Republicans, asserting Wednesday that the Senate immigration measure would boost the U.S. economy and help create jobs.

A Congressional Budget Office report released in June indicated the Senate bill could reduce deficits by $175 billion over the first 10 years and by at least $700 billion in the second decade. The CBO, working with the congressional Joint Committee on Taxation, estimated that eight million unauthorized residents would become legal in the first decade.

In addition, the CBO report estimated the bill would boost the U.S. population by a net of 10.4 million people by 2023 and by 16 million by 2033.

“The benefits are clear. The fact that there is a broad bipartisan consensus behind this is clear,” White House spokesman Jay Carney said Wednesday.

“It cannot be acceptable, broadly and in the long term, that immigration reform would be blocked because some minority of House Republicans is concerned about a primary challenge from the far right. That’s not a good argument. It’s not a good argument politically. It’s certainly not a good argument economically,” he said.

Senate immigration deal includes tougher border security

On the House side, a bipartisan group has been working on an immigration package that differs from the Senate measure by making it harder for undocumented immigrants to get on a path to legal status.

The House measure also would require that border security measures be in place before any process toward residency for undocumented immigrants could begin.

In an interview on Tuesday with PBS, GOP Rep. Trey Gowdy of South Carolina, who chairs a House subcommittee on immigration, made clear that the Senate measure had no chance of passage but also indicated areas of potential compromise regarding a path to legal status.

For example, Gowdy said some immigrants — such as those who served in the military and children of undocumented immigrants brought into the country illegally by their parents — could have a faster mechanism to residency and eventual citizenship.

In addition, he said those who had been in the country illegally for many years and made positive contributions could get a faster path to legal status than illegal newcomers.

In a separate interview, GOP Rep. Raul Labrador of Idaho told MSNBC on Wednesday that the House would take a incremental approach by considering specific provisions that would add up to comprehensive immigration reform in the end.

Like Gowdy, Labrador included legal status for undocumented immigrants as part of the effort.

“It will include a path to legal status. I don’t know that it will include a path to citizenship,” he said. “It will be some sort of legal status, which I think should not prevent anybody from becoming citizens, but will not necessarily give them citizenship status.”

Sticking points in the House debate include whether undocumented immigrants would be eligible for health care benefits during the years it would take to get citizenship, and empowering state and local police to work with federal authorities in enforcing immigration laws.

CNN’s Alan Silverleib, Lesa Jansen, Ashley Killough and CNNMoney’s Jeanne Sahadi contributed to this report.