“We have to make sure that taxes don’t go up on the middle class, that the economy remains strong,” Mr. Obama said as the meeting began.

“That’s an agenda that Democrats and Republicans and independents, people all across the country share,” he said. “So our challenge is to make sure that we are able to cooperate together work together find some common ground, make some tough compromises, build some consensus to do the people’s business.”

“My hope is, is that this is going to be the beginning of a fruitful process where we’re able to come to an agreement that will reduce our deficit in a balanced way, that we will deal with some of these long-term impediments to growth,” he added.

Republican leaders, for their part, have signaled since their losses for the White House and in Congress that they would agree to additional tax revenues, but they oppose raising the top marginal rate, which is now 35 percent and scheduled to return to 39.6 percent, the Clinton-era level, on Jan. 1.

And Republicans are holding out for significant future savings from Medicare and Medicaid, which Mr. Obama and his party also support as necessary to controlling the mounting federal debt in an aging population – to a limit. Still, many liberal Democrats are mobilizing against entitlement program cuts, especially after a campaign in which Republicans and their standard-bearer, Mitt Romney, attacked Mr. Obama and the Democrats for having enacted Medicare reductions as part of the 2010 health-care law.

All four Congressional leaders will remain in their senior posts, both for the lame-duck session that started this week and the new Congress that starts in January. The lame-duck session could last through the holidays unless the parties can reach an agreement on a budget deal to avoid a so-called fiscal cliff after Jan. 1, when the country would be hit by more than $500 billion in tax increases – mainly from the expiration of the Bush tax rates – and across-the-board cuts in military and domestic spending absent a bipartisan compromise on an alternative package of deficit-reduction.

Such large and sudden action could cause a short recession, the Congressional Budget Office has warned; any negotiated alternative would spread the budget cuts over a longer period to avoid harming the still-weak economy.