WASHINGTON (AP) ― President Barack Obama has scaled back his ambitions for a sweeping budget bargain with Republicans. Instead, he’s calling for a limited measure sufficient to prevent the government from careening off the “fiscal cliff” in January by extending tax cuts for most taxpayers and forestalling a painful set of agency budget cuts.

In a White House appearance Friday, Obama also called on Congress to extend jobless benefits for the long-term unemployed that would otherwise be cut off for 2 million people at the end of the year.

Obama’s announcement was a recognition that chances for a larger agreement before year’s end have probably collapsed. It also suggested that any chance for a smaller deal may rest in the Senate, particularly after the collapse of a plan by House Speaker John Boehner, R-Ohio, to permit tax rates to rise on million-dollar-plus incomes.

“In the next few days, I’ve asked leaders of Congress to work toward a package that prevents a tax hike on middle-class Americans, protects unemployment insurance for 2 million Americans, and lays the groundwork for further work on both growth and deficit reduction,” Obama said. “That’s an achievable goal. That can get done in 10 days.”

Maybe, maybe not. The latest plan faces uncertainty at best in the sharply divided Senate. GOP leader Mitch McConnell of Kentucky, who wields great power even in the minority, called Friday for Senate action on a House bill from the summer extending the full menu of Bush-era tax cuts. He promised that it will take GOP votes for anything to clear the Senate, where 60 votes are required to advance most legislation. Democrats control 53 votes.

Boehner, giving the GOP weekly radio address, said, “Of course, hope springs eternal, and I know we have it in us to come together and do the right thing.”

Earlier, Boehner said Obama needs to give more ground to reach an agreement and that both he and Obama had indicated in a Monday telephone call that their latest offers represented their bottom lines. “How we get there,” he added, “God only knows.”

Congress shut down for Christmas and Obama flew to Hawaii with his family for the holidays. But both men indicated they’d be back in Washington, working to beat the fast-approaching Jan. 1 deadline with an agreement between Christmas and New Year’s.

Obama announced his plans after talking by phone with Boehner and meeting with Senate Majority Leader Harry Reid, D-Nev., who had previously pinned his hopes on an Obama-Boehner agreement and is wary of dealing with McConnell.

At the White House, Obama projected optimism despite of weeks of failed negotiations. “Call me a hopeless optimist, but I actually still think we can get it done,” he said.

Boehner spoke in the morning, describing the increasingly tangled attempts to beat the Jan. 1 deadline and head off the perilous combination of across-the-board tax hikes and deep spending cuts.

“Because of the political divide in the country, because of the divide here in Washington, trying to bridge these differences has been difficult,” Boehner said. “If it were easy, I guarantee you this would have been done decades before.”

Obama said that in his negotiations with Boehner, he had offered to meet Republicans halfway when it came to taxes and “more than halfway” toward their target for spending cuts.

It’s clear, however, that there’s great resistance in GOP ranks to forging a bargain with Obama along the lines of a possible agreement that almost seemed at hand just a few days ago: tax hikes at or just over $1 trillion over 10 years, matched by comparable cuts to federal health care programs, Social Security benefits and across federal agency operating budgets.

Obama said he remains committed to working toward a goal of longer-term deficit reduction to reduce chronic trillion-dollar deficits while keeping tax rates in place for nearly everyone.

“Even though Democrats and Republicans are arguing about whether those rates should go up for the wealthiest individuals, all of us ― every single one of us ― agrees that tax rates shouldn’t go up for the other 98 percent of Americans,” Obama said, citing statistics associated with his promise to protect household income under $250,000 from higher tax rates.

Neither the House nor the Senate is expected to meet again until after Christmas. Officials in both parties said there was still time to prevent the changes from kicking in with the new year.

The week began amid optimism that Obama and Boehner had finally begun to significantly narrow their differences. Both were offering a cut in taxes for most Americans, an increase for a relative few, and cuts of roughly $1 trillion in spending over a year. Also included was a provision to scale back future cost-of-living increases for Social Security recipients ― a concession by the president that inflamed many liberals.

GOP officials said some senior Republicans such as Rep. Paul Ryan, R-Wis., the most recent Republican vice presidential nominee, opposed the possible agreement. But No. 2 House Republican Eric Cantor of Virginia has joined arms with Boehner.

Boehner stepped back and announced what he called Plan B, legislation to let tax rates rise on incomes of $1 million or more while preventing increases for all other taxpayers.

Despite statements of confidence, he and his lieutenants decided late Thursday they were not going to be able to secure the votes needed to pass the measure in the face of opposition from conservatives unwilling to violate decades-old party orthodoxy never to raise tax rates.

The retreat came after it became clear that too many Republicans feared “the perception that somebody might accuse them of raising taxes,” Boehner said.

Boehner also said that last Monday he had told Obama he had submitted his bottom-line proposal.

“The president told me that his numbers ― the $1.3 trillion in new revenues, $850 billion in spending cuts ― was his bottom line, that he couldn’t go any further.”

That contradicted remarks by White House press secretary Jay Carney, who said on Thursday that Obama has “never said either in private or in public that this was his final offer. He understands that to reach a deal it would require some further negotiation. There is not much further he could go.”