Sen. Mark Warner, D-Va., said he expects something will be passed, but nothing that will solve the nation's growing financial problems.
"I think there's unfortunately only going to be a small deal," he said, but added "it's critical we get to the big deal."
Obama already has scaled back his ambitions for a sweeping budget bargain. Before leaving the capital on Friday, he called for a limited measure that extends George W. Bush-era tax cuts for most people and stave off federal spending cuts. The president also urged 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.
The failure of Boehner's option in the House has shifted the focus.
"The ball is now clearly with the Senate," said Lieberman.
He said Senate Majority Leader Harry Reid of Nevada and GOP leader Mitch McConnell of Kentucky "have the ability to put this together again and pass something. It won't be a big, grand bargain to take care of the total debt, but they can do some things that will avoid the worst consequences going over the fiscal cliff."
It was only a week ago when news emerged that Obama and Boehner had significantly narrowed 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 scaling back of future cost-of-living increases for Social Security recipients — a concession on the president's part as much as agreeing to higher tax rates was for the speaker.
Lieberman was on CNN's "State of the Union," while Barrasso, Klobuchar and Conrad appeared on "Fox News Sunday." Hutchison and Warner were on CBS' "Face the Nation."