The White House is hoping that public anger at the 16-day government shutdown has so badly damaged the GOP that House Republican leaders will consider immigration reform as a way to improve their popularity with moderate voters.

Obama’s aides also are intent on showing the president is willing to compromise, partly to counter GOP charges that he was inflexible during the bitter shutdown standoff.

In remarks at the White House, Obama hinted that he was no longer tied to the Senate bill, the elaborate product of months of intense bipartisan negotiations, to achieve what he has called a major priority for his second term.

Obama instead signaled that he might consider a package of smaller bills, if necessary, as long as they provide a path to citizenship for the estimated 11 million people in the country without legal status.

“If House Republicans have new and different additional ideas on how we should move forward, then we want to hear them. I’ll be listening,” Obama told several dozen pro-reform activists from labor, business and religious groups.

White House spokesman Jay Carney echoed the shift, telling reporters there are “a variety of ways that you can reach the ultimate goal” of a bill that Obama could sign into law.

“The House’s approach will be up to the House,” Carney said. “There is a comprehensive bill the House Democrats have put together that is similar to the Senate bill and reflects the president’s principles. But the means by which we arrive at our destination is in some ways of course up to the lawmakers who control the houses of Congress.”

The White House effort to resuscitate a bill that seemed all but dead in the House before the shutdown still faces steep and perhaps insurmountable odds. But the jockeying Thursday raised at least some hope that compromise remains possible.

“I hope President Obama meant what he said today about listening to new and different ideas presented by House Republicans,” House Judiciary Committee Chairman Robert W. Goodlatte (R-Va.) said in a statement. “The president should work with Congress, including House Republicans, to achieve immigration reform, and not against us.”

In recent weeks, GOP leaders have worked behind the scenes to craft legislative proposals that might pass muster with rank-and-file Republicans and — if joined with a legalization program — could appeal to the White House.

Majority Leader Eric Cantor and other House Republicans have met in small groups to write bills that would change parts of the immigration system. GOP proposals include adding high-tech visas, revamping farm and low-skilled immigrant labor programs, and ramping up border security.

“I expect us to move forward this year in trying to address reform and what is broken about our system,” Cantor said on the House floor Wednesday.

Whether the House will go as far as the Senate, and include a 13-year pathway to citizenship for qualified immigrants, is far from clear. Republicans seemed unwilling to accept the entire Senate bill, which includes $46 billion over 10 years for extra border security and other programs, as well as numerous legal reforms.

On Thursday, House Speaker John A. Boehner‘s office said the House would not consider “massive, Obamacare-style legislation that no one understands,” referring to the Senate bill. “Instead, the House is committed to a common-sense, step-by-step approach that gives Americans confidence that reform is done the right way.”

In his comments Thursday, Obama offered some unsolicited advice to House Republicans, who took the brunt of the blame for the bruising budget and debt battles of recent weeks.

“Good policy is good politics in this instance,” Obama said. “If folks are really that consumed with the politics of fixing our broken immigration system, they should take a closer look at the polls, because the American people support this.”

Outside analysts and advocates say Obama needs to gain support from House Republicans who might be tempted to support immigration reform but are wary of supporting a bill he has embraced. Simply urging the House to pass the Senate bill may antagonize them.

“He has zero credibility,” said Rep. Mario Diaz-Balart (R-Fla.), who has worked for months on a House bill that would increase border security and make it possible for some immigrants without legal status to pay a penalty and eventually apply for legal status. “If he wants to be helpful on immigration reform, he has to do what he has been doing for the past five years, which is nothing.”

Rep. Luis V. Gutierrez (D-Ill.), who asked the president in a meeting at the White House earlier this year to step back from negotiations in Congress for fear his involvement would spook Republicans, thought Obama struck the right tone Thursday.

“He didn’t say, ‘It’s my way or the highway,’” said Gutierrez, who is involved in discussions with House Republicans on immigration proposals. Gutierrez wants Obama to step up his involvement in crafting a deal, including bringing together both sides for a face-to-face meeting.

“Camp David is a nice place in the fall,” Gutierrez said.

The bigger problem may be time. The House is only in session for five more weeks before the Christmas break. With other business stacked up due to the government shutdown, that leaves scant floor time to debate and pass a complex package of proposals.

Motorola Solutions Inc. Chairman Greg Brown, who heads the Business Roundtable Select Committee on Comprehensive Immigration Reform, saw room for compromise.

“We agree with Speaker Boehner and the president that the time is now to fix our broken immigration system,” he said in a statement. “Our economy needs a boost, and immigration reform will help.”

Frank Sharry, the head of America’s Voice, an immigration reform advocacy group, said he thought the president’s comments “signaled openness.”

Sharry said that a House GOP proposal probably would not be acceptable to Democrats, but “the question is: Is it in the ballpark where both sides can get what they want?”

brian.bennett@latimes.com

christi.parsons@latimes.com