WASHINGTON — Before the bill to end the budget impasse even hit President Obama's desk Wednesday, he and congressional Democrats had pivoted to what they hope is the next big legislative battle: an overhaul of the nation's immigration laws including citizenship for the nation's 12 million undocumented immigrants.

"I look forward to the next venture, which is making sure we do immigration reform," Senate Majority Leader Harry Reid, D-Nev., said late Wednesday.

"Good luck," said Rep. Trey Gowdy, R-S.C., who chairs the House immigration committee.

House Republicans emerged from the 16-day shutdown fight angered at the White House and Reid for refusing to negotiate over the terms to end the shutdown and lift the nation's debt ceiling, and emboldened to wage more strategic legislative strikes in the future.

Gowdy has been moderately supportive of immigration changes and says he maintains good relationships with House Democrats. "But it's a little disingenuous to treat the House as an irrelevant branch of government and then say, 'By the way, tomorrow you'll need to go ahead and push (immigration reform),'" Gowdy said. "It doesn't work that way."

Democrats are hopeful that Republicans, mindful of the party's poor standing nationally with Hispanics, will support a comprehensive approach. Obama said Thursday that immigration is one of the three agenda items he wants Congress to pass this year. Sen. Charles Schumer, D-N.Y., told MSNBC Thursday that an immigration overhaul is "the thing (Obama) wants to get more than anything else" in his second term.

House Speaker John Boehner, R-Ohio, remains publicly and privately committed to advancing immigration legislation in this Congress, but there is virtually no interest among GOP lawmakers to advance the kind of sweeping bill that Democrats are seeking. Instead, Republicans are more likely to pursue a piecemeal approach to address issues individually, such as border security and visas for high-skilled workers.

Rep. Raul Labrador, R-Idaho, a Tea Party conservative who was once a member of a bipartisan House group that tried to draft a broad immigration bill, said the prospects for even smaller bills are slim in the House.

"It's not going to happen this year," Labrador said. "After the way the president acted over the last two or three weeks where he would refuse to talk to the speaker of the House ... they're not going to get immigration reform. That's done."

Getting immigration changes through the House was always going to be a difficult task. The majority of House Republicans have consistently opposed the bill passed by the Senate in July that allows the nation's undocumented immigrants to apply for U.S. citizenship after 13 years, something many in the conference refer to as "amnesty."

That means Boehner, who struggled to unify his members throughout the shutdown, would have to "divide the conference" to pass an immigration bill, said Rep. Tim Huelskamp, R-Kan.

"That would really melt down the conference," said Huelskamp, a Tea Party conservative.

That faction of the House wielded considerable influence over the chamber throughout the 16-day shutdown, pushing Boehner to demand cuts or delays to the president's signature health care law. Those conservatives were supported by Sen. Ted Cruz, R-Texas, a presidential contender for 2016 whose saw his notoriety grow as he railed against Obamacare and the national debt.

And while Cruz was marginalized earlier this year during Senate hearings on the immigration law, his opposition to the bill may get new life through the more conservative wing of the House.

"As we've recently seen, he has the ear of a number of people in the House and I think he's going to be a factor," said Rep. Mario Diaz-Balart, R-Fla., another member of the bipartisan House immigration group who remains confident something can pass.

Democrats in the House saw something else going on Wednesday.

In the end, the Affordable Care Act was not delayed or dismantled and House Republicans agreed to raise the nation's $16.7 trillion debt ceiling.

"You could hear the hissing sound of the pent-up, perceived power being relieved," said Rep. Joe Garcia, D-Fla., who filed the House version of a sweeping immigration bill that has now garnered 182 Democratic co-sponsors.

Rep. Luis Gutierrez, D-Ill., one of the main proponents of getting an immigration bill through Congress, is looking to history for signs of optimism that the House can pass something.

Gutierrez was in the House during the last government shutdown in 1996, and he says Republicans emerged from the damaging closure scurrying to pass "big things" to show the country they could get things done. In the aftermath of that shutdown, the government passed welfare reform, the sweeping Kennedy-Kassebaum health care law and an increase in the minimum wage.

"It was in people's self-interest to pass some good stuff," Gutierrez said. "That's what's going to drive a lot of what goes on around here."

Rep. Xavier Becerra, R-Calif., chairman of the House Democratic Caucus, looks to more recent examples for hope. He said Boehner has violated the so-called 'Hastert Rule' — requiring support from a majority of the majority party in the House before a bill can come to the floor — on several votes that were critical, including emergency relief for victims of Superstorm Sandy.

Becerra sees a similar situation developing on immigration, where the vast majority of Democrats and a small number of Republicans could pass a bill through the House.

"Once again, the speaker for the majority party is going to be placed in a position of deciding whether he's going to put country before party and get something done," Becerra said. "We just need a few courageous Republicans to stand up and say they're ready."

Contributing: Susan Davis