Congress left town Thursday without a deal to avoid a government shutdown in the new year, but lead negotiators for both parties said that they will continue to work over the Thanksgiving break and that they are optimistic about reaching an agreement.

“I’m hopeful,” said House Budget Committee Chairman Paul Ryan (R-Wis.).

“Chairman Ryan and I are working closely together to find a path forward in good faith,” added Senate Budget Committee Chairman Patty Murray (D-Wash.).

Neither Ryan nor Murray would comment on the status of the talks, which have focused on identifying alternative savings to replace sharp agency budget cuts known as the sequester. Absent an agreement, the government will shut down when the current temporary funding measure expires on Jan. 15.

Both sides are eager to avoid another shutdown, particularly Republicans, who suffered greatly in public opinion polls after closing agencies for 16 days in October.

As a result, people close to the talks said Murray and Ryan have agreed to jettison for now any discussion of tax increases or cuts to federal health benefits — each side’s hot button — and focus instead on a more politically neutral category of potential savings known as “other mandatories.”

“That’s how you reach agreement: Stay away from each other’s stoves,” said Rep. Tom Price (R-Ga.), a member of the budget conference committee led by Ryan and Murray and formally tasked with devising a budget plan.

The list of potential “other mandatory” savings — cuts to mandatory programs other than Social Security and Medicare — has been around since at least 2011, when Vice President Biden and House Majority Leader Eric Cantor (R-Va.) attempted to reach a budget deal.

It contains some spending reductions, including in farm subsidies that both parties support, and smaller contributions to civil employee pensions. But most other items would raise money for the Treasury through, for example, broadband spectrum auctions, increased aviation fees and higher fees for companies whose private pensions are insured by the federal government.

Although the category does not include outright tax increases, which Democrats are seeking, or cuts to Medicare benefits, which Republicans have demanded, it does include some politically touchy items.

Democrats, for example, are reluctant to reduce benefits again for federal workers, who already have faced furloughs and a pay freeze. House Speaker John A. Boehner (R-Ohio) does not want budget negotiators to claim any savings from farm programs or food stamps, arguing that those savings already have been approved — and therefore banked — by the House. And neither side is eager to wade into Tricare for Life, the Medicare supplement that is available to military retirees at a very low cost.

That leaves Ryan and Murray struggling to amass enough savings to provide meaningful sequester relief. People close to the talks said an agreement would include no more than $100 billion — and could easily be as low as $40 billion. That would be enough to stop a $20 billion sequester reduction set to hit the Pentagon in January and provide an equal amount of relief to domestic agencies in fiscal 2014.

A deal that small might satisfy defense hawks. But it would infuriate appropriators in both parties, who are hoping that Ryan and Murray will identify enough savings to roll back a portion of the sequester for fiscal 2015 as well.

“That’s very critical,” said House Appropriations Committee Chairman Harold Rogers (R-Ky.), who said he hopes the two will find a way to replace the entire annual cost of the sequester — about $109 billion — for both 2014 and 2015.

“That would give us a chance to get this train back on track” by passing full appropriations bills for federal agencies, Rogers said. “If we don’t get a 2015 number, we’re back in this business next year, another crisis-to-crisis, shutdown mode, which we’re desperately trying to get away from.”

It seemed unlikely, however, that Ryan and Murray were aiming for a deal big enough to replace the entire sequester. After two public hearings of the entire budget conference committee, the pair have entered private talks and are holding details close.

And after Ryan briefed Senate Republicans on Wednesday, many emerged with major doubts about whether a deal could be struck at all.

“No one’s walking out of that place making big bets,” said Sen. Roy Blunt (R-Mo.).

On Thursday, Boehner said that if the effort fails, he will move quickly to pass a temporary measure to keep the government open while maintaining the sequester, despite complaints in his party about the coming hit to the Pentagon.

That puts the speaker on the same page as Senate Minority Leader Mitch McConnell (R-Ky.). McConnell has spent much of the past week urging Republicans not to do anything to undermine the sequester, which has helped to push federal spending down for two years in a row — the first time that has happened since the end of the Korean War.

“I think it’s a bad idea to revisit a law that is actually working and reducing spending for the government,” McConnell told reporters. “Within those constraints,” he said of Ryan and Murray, “I wish them well.”