WASHINGTON — 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.”