The House is expected to approve a continuing resolution this week that would keep the government funded until April 8, with the Senate expected to follow suit. Negotiations will then continue on another short-term spending bill.
But lawmakers said a much larger challenge looms to address the nation's bleak financial outlook, as they must soon decide whether to raise the nation's debt limit.
Senate Minority Leader Mitch McConnell, R-Ky., vowed Sunday that Senate Republicans will not vote to raise the debt limit without a "credible" plan to address the country's $14 trillion debt.
"Democrats can raise it themselves if they choose to and try to do nothing whatsoever about the problem," McConnell said on Fox News Sunday. "I think to get any of the 47 Republicans, you've got to do something... that the markets believe is credible, that the American people believe is credible, that foreign countries believe is credible, in addition to simply raising the debt ceiling."
The challenge of coming up with a bipartisan plan to bring down the debt has fallen to a group led by Sens. Saxby Chambliss, R-Ga., and Mark Warner, D-Va.
"If we don't get our arms around it now, then we're going to become a second-tier nation, and we can't allow that to happen," Chambliss said on Fox.
The group is considering multiple options, including cutting entitlements and recommending that the retirement age for Social Security be raised, said Warner, who also spoke on the program.
"You've got to put everything out," Warner said. "We didn't get into this situation overnight. We're not going to dig out overnight."
"Folks under 35 might see a slight bump in their age increase, but frankly, a lot of folks under 35 don't think there's even going to be Social Security if we don't do something," he said. He added the goal is to make Social Security stable for decades to come.
Chambliss said the proposal the group is developing aims to reduce the effective tax rate across the board by making changes to the tax code. He did not specify what those changes entail.
But he said a key issue for the group is ensuring that Congress does not turn around and spend revenue raised or saved by the plan.
"Frankly, one of the major issues that we are dialoguing about in our group now is what do we do with those revenues," Chambliss said. "We need to make sure that we commit the most significant part of those revenues to tax reduction -- tax rate reduction. Get our corporate rate down to where we're competitive in the world marketplace. Get our individual rates down to where people actually do pay less in taxes."
He said a portion of the revenue would then be used to pay down the nation's debt.
Exactly when the group will roll out a final plan is unclear. "We want to make sure we get it right more than set an arbitrary timeline," Warner said.
He said he would be worried about trying to tie the group's plan to the upcoming vote on raising the debt ceiling. But he added that waiting until next year would be problematic because it's an election year.
"We may not have that long of time before the financial markets say we're going to either no longer want to buy American debt or charge such a higher interest rate on it that it would have a dramatic negative on the economy," Warner said.
In the interim, there has been some criticism that President Obama has not been involved enough in budget negotiations. Majority Whip Dick Durbin, D-Ill., disputed such claims in an interview on CNN's State of the Union.
"I think there's a perception and a frustration among members of Congress that things aren't moving to a conclusion. The president is working behind the scenes," Durbin said.
"I've met with him with leadership," Durbin added. "I know he is reaching out to try to find some accommodation here. He is trying to reach a point where we acknowledge the obvious. We have a serious deficit problem, borrowing 40 cents for every dollar we spend."
Senate Minority Whip Jon Kyl, in turn, rebutted that. The Arizona Republican said, "He may have been talking to Senator Durbin and other Democrats, but he has not been talking to Republicans."
Notably, McConnell said Senate Republicans will not filibuster or block unrelated spending bills from coming to the Senate floor. Instead, he said Republicans would seek to add amendments to those bills cutting funding.