Now that Congress voted to end the shutdown, President Obama is stepping up pressure on his nemesis – House Republicans – to move the ball forward on comprehensive immigration reform.

On Thursday, Obama reiterated the theme he’s been pushing more aggressively in recent days – after the shutdown’s end, Congress needs to make immigration reform a top priority and finish a bill by the end of the year.  

A bipartisan bill passed in June, but efforts to pass a measure on the matter in the House have stalled.

“There are things that we know will help strengthen our economy that we could get done before this year is out,” Obama said in a statement Wednesday night. “We still need to pass a law to fix our broken immigration system. We still need to pass a farm bill. And with the shutdown behind us and budget committees forming, we now have an opportunity to focus on a sensible budget that is responsible, that is fair, and that helps hardworking people all across this country.”

Conservative Republicans in the House, where the GOP has a majority, have vowed not to pass a measure that would give undocumented immigrants a path to legalization, which they say amounts to amnesty. 

But proponents of a path to legal status – which was a key component of the Senate bill – say that it is the only realistic solution to bringing undocumented immigrants onto the radar.

Earlier this week, Obama said to the Los Angeles affiliate of Spanish-language television network Univision: “Once that's done [the budget deal], you know, the day after, I'm going to be pushing to say, call a vote on immigration reform.”

The president's domestic agenda has been sidetracked in his second term by one problem after another.

As he coped with the revelation of domestic surveillance programs, chemical weapons in Syria, and a fiscal battle that has shut down the U.S. government and threatens a debt default, immigration has been relegated to the back burner.

But Obama, who won re-election with overwhelming Hispanic backing, had hoped to make reforms easing the plight of the 11 million immigrants who are in the United States illegally.

Meanwhile, Rep. Raul Labrador, a Republican from Idaho, expressed doubt that Republicans and Democrats could work together on the emotional issue of immigration after an acrimonious experience with the shutdown and debt ceiling.

“Absolutely not,” Labrador said at a conservative panel on Wednesday, according to Politico. “If the president is going to show the same kind of good faith effort that he’s shown in the last couple of weeks, I think it would be crazy for the House Republican leadership to enter into negotiations with him on immigration.”

Labrador was part of the so-called Group of Eight in the House, which consisted of four Democrats and four Republicans who were working on an immigration bill. But Labrador left the group in the summer, amid differences with the group. Two other GOP members also left, leaving only Rep. Mario Diaz-Balart of Florida.

“I think what [President Barack Obama] has done over the last two and half weeks, [is] he’s trying to destroy the Republican Party and I think that anything we negotiate right now with the president on immigration will be with that same goal in mind, which is to destroy the Republican Party and not to get good policies,” Labrador said.

A spokeswoman for the Republican National Committee in Washington D.C. balked at Obama’s statements saying he is committed to seeing an immigration bill this year.

“We’ve seen this type or rhetoric before from the President,” said Izzy Santa, the RNC spokeswoman. “Republicans in the House have been working on immigration. In the Senate, Marco Rubio led immigration reform. If Democrats were so concerned with the issue, they would have sidestepped the President and done something on the issue, and they didn’t. They’re just pandering.”

Immigration advocates, who have expressed as much frustration over President Obama as they have members of Congress, and have blamed all of them for inaction on immigration reform, said they too are growing skeptical. Nearly 2 million people -- a record -- have been deported during Obama's tenure. 

One group, America’s Voice, said in a statement that Republicans have a chance now to move on immigration.

“The central question is whether Speaker Boehner and smart Republicans in the House will take the get-out-of-jail card that Democrats are offering to them on immigration reform,” said Frank Sharry, who heads America’s Voice, a group that advocates for a path to legal status for undocumented immigrants.

“Working with Democrats to pass reform will help the GOP rehabilitate their badly damaged brand," Sharry said, "solve a huge political problem facing the GOP with respect to Latino, Asian and immigrant voters, and prove to the American people they can govern responsibly rather than recklessly. “

Reuters contributed to this report.

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Elizabeth Llorente can be reached elizabeth.llorente@foxnewslatino.com

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