After the meeting, Oklahoma Rep. Tom Cole told reporters that Boehner said: "I've stayed out of those negotiations."

"Every time we get involved in them, we sort of get burned, so we're going to let the Senate work its will, see what they do and what they send us, and we'll act accordingly," he said.

Sen. Chuck Schumer, D-New York, told ABC's "This Week" he thought the chances of a short-term, last-minute deal brokered by Senate leaders were better than 50-50.

"I've been a legislator for 37 years, and I've watched how these things work on these big, big agreements," Schumer said. "They almost always happen at the last minute."

And Sen. Lindsey Graham, R-South Carolina, said the chances are "exceedingly good" that some type of deal will be reached by Monday night.

"I think, whatever we accomplish, political victory to the president, hats off to the president. He stood his ground. He's going to get tax rate increases, maybe not $250,000, but upper-income Americans," Graham said on "Fox News Sunday."

"And the sad news for the country is that we have accomplished little in terms of not becoming Greece or getting out of debt."

Other Republicans argued Sunday that Obama's plan hasn't done enough to limit spending.

"The president is doing nothing about the addiction that his administration has to spending. He's the spender in chief," Sen. John Barrasso of Wyoming said on CNN's "State of the Union."

Obama told NBC that he has cut more than $1 trillion in spending and offered another $1 trillion-plus in additional cuts "so that we would have $2 of spending cuts for every $1 of increased revenue." He said the majority of Americans have made clear they support his calls for "a balanced approach" that would increase taxes on the wealthy.

Sen. Dick Durbin, D-Illinois, warned that now is "exactly the wrong time to go over this cliff."

"We are in the midst of an economic recovery. We are seeing new job creation, businesses are seeing new growth, we are seeing the kind of economic indicators we've been waiting for for years. This going over the cliff is going to bring uncertainty to our markets, and with that uncertainty, a pullback in consumer confidence and a reduction, I'm afraid, of business activity and the creation of new jobs," he said.

But former Democratic Party chairman and 2004 presidential contender Howard Dean told ABC that heading over the cliff would not be so bad, calling it a "fiscal curb" instead.

"You go back to the Clinton tax rates, and you make some significant cuts. And you cut the Defense Department, which hasn't been cut in 30 years," said Dean, a former Vermont governor. Meanwhile, going over the cliff gives Obama more leverage, "because then all of sudden, middle-class people's taxes are going to rise, and that's going to be bad for every politician in Washington."

"Maybe they'll actually get something done," he said. "But I think, at this point, at this late hour, I think almost any deal they come up with is worse than going over the cliff."

Sen. Olympia Snowe, R-Maine, told CNN's "State of the Union" that she expects Congress will vote to extend tax cuts for incomes below $250,000 -- perhaps below $400,000 -- before midnight Monday. A Senate agreement "would build momentum" for the move in the House, she said.

"I think it would be horrific for the country if at this time, the final days of this legislative session that already has reached historic proportions of failure, that we would now culminate in failure to extend these tax cuts," said Snowe, who is on her way out of office.