Many Republicans say they want a permanent extension of the rates, or as long an extension as possible. Democrats say they would not mind the issue coming up during Mr. Obama’s re-election bid, because they see it as politically helpful to them in painting Republicans as defenders of the rich. The debate, of course, could cut the other way, with Republicans again portraying Democrats as seeking to raise taxes.

Some Congressional Democrats have suggested a willingness to let the tax rates expire for everyone at the end of the year, and to press the fight into next year. At the same time, some prominent economists have counseled that it would be wise not to substantially raise taxes or cut spending at the moment given the risk of further hindering an already precarious economic recovery.

In an interview on “60 Minutes” on Sunday night, the chairman of the Federal Reserve, Ben S. Bernanke, said policy makers should be careful to protect the recovery in what could be interpreted as a warning to Democrats not to raise taxes and to Republicans not to cut government spending. “We don’t want to take actions this year that will affect this year’s spending and this year’s taxes in a way that will hurt the recovery,” Mr. Bernanke said. “That’s important.”

But Mr. Bernanke also warned about the dangers of the nation’s fast-growing debt, which will be increased by $4 trillion or more over 10 years as a result of the tax deal being worked out by Republicans and the White House, and said policy makers must start thinking now about the “long-term structural deficit.”

After the failed votes in the Senate on Saturday, top Democratic Congressional leaders met at the White House with Mr. Obama, who told them he would not agree to any deal unless it included the extension of jobless aid, which has begun to run out, and also the extension of a number of tax breaks for middle- and lower-income Americans that were included in last year’s economic stimulus plan.

In a twist on the Republicans’ argument that letting the Bush-era tax cuts expire would amount to a tax increase in a weak economy, Mr. Obama said allowing the stimulus provisions to expire would result in a similar tax increase on working-class Americans who could afford it even less than the high-income people who would benefit from the Republicans’ plan.

The Democratic Congressional leaders attended another strategy session at the residence of Vice President Joseph R. Biden Jr., who has taken a personal hand in the negotiations with Senate Republicans, with whom he has personal relationships from his long tenure as a senator.