"It's always been an uphill battle," White House spokesman Jay Carney told reporters. "Hard things are hard."
Carney said the White House and other supporters of the sweeping bipartisan immigration bill that passed the Senate "can't prevent [House] lawmakers from making bad choices."
"We can simply be part of a broad, comprehensive effort of persuasion that makes a case for why passing comprehensive immigration reform is the right thing to do for the economy, for our businesses, for the middle class," Carney said. "It's the moral thing to do, and we'll keep at it and we'll get it done."
But a senior Democratic aide told The Hill, "If I were betting on this, I'd say it's not looking good -- not at all.
"If anything passes it's going to be some watered-down bill beefing up border security agents or something along those lines," the aide said.
House Speaker John Boehner, R-Ohio, and others GOP leaders said in a statement after a closed-door meeting Wednesday, "Today House Republicans affirmed that rather than take up the flawed legislation rushed through the Senate, House committees will continue their work on a step-by-step, common-sense approach to fixing what has long been a broken system."
"We need to do it in a very thorough way, we want to do it the right way," said House Budget Committee Chairman Paul Ryan, R-Wis., who has supported an immigration overhaul, provided the border with Mexico is "secured."
"We don't want to rush anything," Ryan said after the meeting, before walking into an elevator.
In the meeting, Boehner and Ryan urged their colleagues to pass immigration reform legislation, with Boehner arguing Republican lawmakers would "be in a much weaker position" if they don't act, Rep. Tom Cole, R-Okla., told The Hill.
Boehner and other House GOP leaders are reaching out to top House Democrats to see if they'd support a piecemeal immigration-reform approach, Politico reported.
Former President George W. Bush, a Republican who supports a path to citizenship for the 11 million immigrants living in the United States illegally, called for a "positive resolution" to the immigration debate.
"I hope during the debate that we keep a benevolent spirit in mind and we understand the contributions immigrants make to our country," Bush, who was not able to remake the immigration system when he was president, said at a naturalization ceremony in Dallas.
But Rep. Tim Huelskamp, R-Kan., a member of the Tea Party Caucus, said immigration reform is "actually the No. 10 issue on a Top 10 list of issues" of his constituents, The Wall Street Journal reported.
The path to citizenship is a big stumbling block, GOP lawmakers were quoted by the Journal as saying.
"I am not here to reward people for illegal behavior," said Rep. Paul Gosar, R-Ariz. "Everyone worries they're in the shadows -- well, they chose to be in the shadows by doing this illegally."
"Citizenship is not on the table -- it's a non-starter," Rep. Matt Salmon, R-Ariz., said.
But Rep. Spencer Bachus, R-Ala., said he supported a pathway to citizenship "because I don't believe we should have a second class of citizens."
Everyone living in the United States should feel invested in the country, he said. Denying that would create "an underclass and I don't believe that's what America is all about."