WASHINGTON (Reuters) - Treasury Secretary Timothy Geithner pushed Republicans on Sunday to offer specific ideas to cut the deficit, and predicted that they would agree to raise tax rates on the rich to obtain a year-end deal and avoid possible economic doom.

But the top U.S. Republican, Speaker of the U.S. House of Representatives John Boehner, stood firm and renewed his stand against increased tax rates, leaving talks at a stalemate.

"Here's the problem," Boehner told "Fox News Sunday" as both sides took their battle to TV talks shows. "When you go and increase rates, you make it more difficult for our economy to grow," he said.

Besides, Boehner said, if Republicans agreed to give President Barack Obama $1.6 trillion in new tax revenue, "He's going to spend it," not reduce the deficit.

Geithner, Obama's top negotiator, said in a separate appearance on Fox that Republicans must step up.

The treasury secretary said Republicans will be responsible if no deal is reached by the end of the month, triggering the "fiscal cliff," deep automatic spending cuts and across-the-board tax hikes that could plunge the country into a recession.

"There's not going to be an agreement without rates heading up," Geithner said on CNN's "State of the Union."

With polls showing most Americans favor raising tax rates on the wealthy and cracks starting to appear in what had been a solid wall of Republican opposition to such a move, the Obama administration figures it has the upper hand.

But Boehner made it clear that the fight has only begun and he is interested in cutting a deal, not sounding fiscal alarms. "I don't want any part of going over the cliff. I'm going to do everything I can to avert that," Boehner said.

He again refused to offer specific deficit reduction proposals, other than to repeat that one option would be to end a number of unspecified tax deductions.

"The president has seen a lot of options from us. There are a lot of them on the table and I'm hopeful that the conversation will continue," Boehner said.

Boehner also reaffirmed his party's opposition to Congress giving the president sole authority to increase the U.S. debt limit, a power both Democrats and Republicans value.

"Silliness. Congress is never going to give up this power," Boehner said, explaining it provides lawmakers needed leverage in dealing with the White House.

'FLABBERGASTED'

More talks are expected this week, at least at the staff level. But both sides have said it may be another week or so before negotiations get serious.

A likely scenario is a possible short-term fix that would postpone the deadline for the fiscal cliff for six months to a year. Geithner's opening offer last week included raising tax revenues by $1.6 trillion, at least $50 billion in new economic stimulus spending and effectively giving the president the ability to raise the debt limit.

Republicans promptly rejected the offer as unacceptable, even laughable.

Boehner said he was "flabbergasted" and recalled telling Geithner, "You can't be serious."

Also on Sunday, the top House Democrat, Nancy Pelosi, renewed her threat to force a vote on a Senate-passed plan to extend tax cuts for the middle class if Boehner does not schedule a vote "immediately."

"The clock is ticking and stalemates are a luxury we cannot afford," she said in a written statement.

If Congress does not act, the temporary tax cuts for all taxpayers enacted under former President George W. Bush will expire on December 31. Republicans want to extend those tax cuts for all taxpayers, while President Barack Obama and his fellow Democrats want the tax cut extended only for those with incomes under $250,000.

Republicans, who control the House but are the minority in the Senate, have expressed a willingness to raise revenues by such steps as limiting tax deductions, but most oppose increasing rates.

The combination tax hikes and spending cuts set to take hold early next year would suck about $600 billion out of the economy.