Mr. Obama raised the issue in his first comments after lawmakers reached a deal to reopen the government, and on the night the shutdown ended the three top Senate Democrats said they hoped to extend the bipartisan moment that produced the compromise by taking up immigration.

“Let’s move on,” said the Senate majority leader, Harry Reid of Nevada. He added that he hoped “the next venture is making sure we do immigration reform.”

But the possibilities for progress on the issue will be determined in the House of Representatives, where many conservative Republicans are fuming with frustration over their meager gains from the two-week shutdown and turning their ire against Mr. Obama, saying he failed to negotiate with them. It will be up to Speaker John A. Boehner of Ohio to discern whether relations with the White House are simply too raw for House Republicans to consider legislation on an issue the president has made a priority.

Many Democrats and some Republicans are arguing that passing a broad immigration bill could be a way for Republicans to come back from the bruising they took in the polls during the shutdown. “When the Republican polling numbers are at 20 percent, there’s a pretty strong argument to do something to get those poll numbers up, and immigration is a good way to do that,” said Senator Charles E. Schumer, Democrat of New York.

The effort to repair the immigration system has attracted support across the political spectrum, including from traditional Republican allies like business, agriculture and evangelical Christians. The Senate passed a sweeping bill in June on a bipartisan vote. But lawmakers on both sides agreed that the window for action on immigration is narrow, most likely limited to the remaining months of this year, before the next fiscal deadlines, or maybe to early next spring.

Mr. Boehner would like to make progress this year on immigration, a spokesman said Friday. “The speaker remains committed to a step-by-step process to fix our broken immigration system,” said the spokesman, Michael Steel.

But House Republican leaders are waiting to gauge the intensity of distrust of the White House among fractious conservatives in their caucus.

“The president’s attitude and actions over the past few weeks have certainly made getting anything done on immigration considerably harder,” a senior Republican aide said.

Several Republicans who had been working on immigration bills before the shutdown said they no longer believed Mr. Obama would negotiate fairly.

“I think what he has done over the last two and a half weeks, he’s tried to destroy the Republican Party,” Representative Raúl R. Labrador, Republican of Idaho, said Wednesday. “I think that anything we do right now with this president on immigration will be with that same goal in mind, which is to destroy the Republican Party and not to get good policies.”

If the House takes any action, it will probably be on smaller measures that emerged earlier this year from the Judiciary and Homeland Security Committees. The Judiciary Committee, led by Representative Robert W. Goodlatte of Virginia, approved four bills that deal with enforcement, agricultural guest workers and high-skilled visas. A homeland security bill would tighten border security.

Republican lawmakers, including Mr. Goodlatte and Representative Eric Cantor of Virginia, the House majority leader, have been working on a bill to give legal status to young immigrants who came here illegally as children. But no bill has been introduced and no decisions have been made on who would introduce it, Republican aides said Friday.

House Democrats predicted Republicans would be spurred on by an interest in showing skeptical voters that they can produce results.

“I’m more optimistic because what prevailed at the end of this Republican shutdown and potential default was a clear message that this is not how you run government,” said Representative Xavier Becerra of California, the chairman of the House Democratic Caucus.

Republicans who want an immigration bill were encouraged that Mr. Boehner survived the shutdown retaining the support of the conservatives who set it off; they hope the speaker will be emboldened to move on immigration legislation, which is still controversial within his conference.

Many Democrats say Mr. Boehner could quickly pass a bill similar to the Senate’s if he would allow a vote like the one that ended the shutdown, to pass legislation with both Democratic and Republican votes.

Pressure for that option from advocacy groups is growing and intensifying. In recent days, protesters held sit-ins and blocked roadways at immigration detention centers in several cities.

“For our families, this is intensely personal,” Kica Matos of the Fair Immigration Reform Movement, a rights coalition, said Friday. “It’s about keeping mothers with sons, fathers with daughters. We will no longer stand for the Republican family separation agenda that has torn us apart for far too long.”