House Speaker John Boehner, right, meets with other members of the Congressional leadership Wednesday to fine-tune the details of a potential budget deal. (Photo by Bill O'Leary/The Washington Post) (Bill O'Leary/TWP)

A round of successful deal-making on Capitol Hill has altered the political dynamic in Washington, raising hopes within the Obama administration that stalled second-term priorities such as immigration might still have a chance at success.

The key movement has come in the House, where members of both parties overwhelmingly approved a two-year budget deal and where Speaker John A. Boehner (R-Ohio) declared war on the tea party groups that have bedeviled him.

The developments heighten the possibility of progress on issues such as immigration, which has been at the top of President Obama’s agenda since he won more than 70 percent of the Latino and Asian vote on the way to reelection last year.

The budget agreement, which is expected to pass the Senate, also paves the way for a more normal appropriations process that will give the White House additional flexibility in deciding where to spend a bigger pot of money over the next two years. That could allow it to restore funding for priorities such as Head Start and the National Institutes of Health, although Republicans will have a say.

Administration officials hailed the budget agreement Friday as an important step toward lifting the threat of another government shutdown like the one in October, which dominated the debate in Washington at the expense of other issues.

The White House also signaled that it would seize the moment by trying to improve rocky relations with congressional leaders, announcing Friday that it had hired a longtime Capitol Hill staffer, Katie Beirne Fallon, as its new legislative affairs chief.

“Expectations have, for good reason, been set pretty low when it comes to compromise in Congress, but a budget deal . . . represents just that,” White House press secretary Jay Carney said. “And it’s not a huge step, but an important step forward in what we believe can be done if both sides address these challenges that the American people want us to address in a way that acknowledges that compromise means you’re not going to get everything you want.”

At the same time, it’s not clear whether staunch GOP opposition to Obama’s agenda — something the president has referred to as a “fever” — will abate enough to allow him to move forward with other priorities. Obama also remains hobbled by sagging approval ratings and the botched rollout of his health-care program.

The administration got an unexpected boost this week when Boehner berated the tea party-aligned groups that have caused him, and Obama, so many headaches since 2010. He said the groups, which have fiercely opposed the budget deal and other bipartisan agreements, are “misleading” and lack credibility.

Carney said Friday that he would not make “assessments of internal Republican Party dynamics.” But the sense that a political reshuffling was afoot was reflected by the reaction from the outside groups that Boehner had criticized.

Michael Needham, chief executive of Heritage Action, said in a Friday interview on MSNBC that Boehner’s goal in picking a fight with a wing of the party that has never fully embraced his leadership was to distract from the flaws of the budget deal and clear a path for a compromise on immigration.

Boehner wants to “cast aside the conservative elements in pursuit of policies that part of the party doesn’t like,” Heritage Action communications director Dan Holler said. “It’s all aligning in that direction.”

On the budget, the Democratic-controlled Senate appears to be on track to follow the lead of the GOP-led House, with a final vote expected late next week.

There is some opposition to the pact among Senate Republicans, who were largely left out of the negotiations and are also angry about Democratic rule changes that have made it easier to approve Obama administration nominees. That means the Senate tally on the budget is likely to be narrower than in the House, which approved the deal by a ratio of 3 to 1 late Thursday.

“I am confident that we will have a bipartisan support because I believe the American people very strongly want us to get back to a place where we are able to govern and start restoring their confidence. I’m confident we’ll get there,” Senate Budget Committee Chairman Patty Murray (D-Wash.) told reporters Friday.

Nonetheless, the budget agreement is limited and offers no clear path for many of Obama’s biggest priorities, including raising the minimum wage, funding early-childhood education and investing in infrastructure. Such major expenditures likely had their best chance to be accomplished as part of a big budget process. But with the process set to be wrapped up by Jan. 15, there likely won’t be enough time.

That’s why much of the focus is on immigration. The Senate approved a comprehensive plan in June featuring a path to citizenship for the nation’s estimated 11 million to 12 million illegal immigrants, but House Republicans shelved that deal and vowed to focus on smaller-scale bills instead.

The issue has languished since then as Congress became consumed with debates over fiscal matters and the Obama administration scrambled to fix its health-care rollout.

Republicans who support immigration say the party would be foolish not to move forward with its own plans to overhaul the nation’s immigration laws. As Obama’s approval ratings have fallen, Hispanic support has dropped off at a steeper rate than among other groups, according to a Gallup poll this month, giving the GOP an opportunity to make inroads with a fast-growing segment of the electorate.

“There are members who say, ‘Latinos always vote Democrat,’ but I always say go with the polls,” said Alfonso Aguilar, executive director of the Latino Partnership for Conservative Principles. He pointed to a February Gallup poll showing that 50 percent of Latinos identify as independent. “When pushed, they said they’d likely vote Democratic, but that’s because Republicans are not giving them an alternative. Why do they have that reaction to Republicans? It’s because of immigration,” he said.

Talk of a revived immigration debate is not the only evidence of a possible change in atmosphere in Washington. There’s also the matter of holiday plans: The White House announced that Obama, who has delayed his winter break for three years running because of fiscal fights, will leave for Hawaii as scheduled next weekend.

Zachary A. Goldfarb, Ed O’Keefe and Paul Kane contributed to this report.