Speaking at the Women in Finance Symposium in Washington, Geithner said: "Let me be clear: The debt limit will be raised. Failure is not an option. Both sides understand what is at stake."
He said to Republicans: "You can't ask middle-class families and retirees to bear the entire sacrifice ... on their own."
At the White House, press secretary Jay Carney said it is not up to President Barack Obama alone to propose a deal.
"The president doesn't have a vote in this. It's Congress that has to act," Carney said.
He refused to say whether Obama is willing to raise the retirement age for Medicare and Social Security.
Tuesday's negotiating session lasted about an hour and 45 minutes in the White House Cabinet Room, and included House Speaker John Boehner, R-Ohio, Senate Majority Leader Harry Reid, D-Nev., Senate Minority Leader Mitch McConnell, R-Ky., House Minority Leader Nancy Pelosi, D-Calif., House Majority Leader Eric Cantor, R-Va., Senate Majority Whip Dick Durbin, D-Ill., Senate Republican Whip Jon Kyl, R-Ariz., and House Democratic Whip Steny Hoyer, D-Md.
At a news conference Monday before heading to the White House for a meeting, Boehner said the only Republican concession Obama should expect is a vote raising the federal debt limit itself.
"Most Americans would say that a 'balanced' approach is a simple one -- the administration gets its debt-limit increase and the American people get their spending cuts and their reforms," he said. "And adding tax increases to the equation doesn't 'balance' anything."
Obama often describes his budget-discussion desire as seeking a "balanced approach" to deficit reduction.
At a news conference before Monday's talks, Obama repeated his desire and said he was "prepared to take on significant heat from my party to get something done -- and I expect the other side should be willing to do the same thing."
He said unless Republicans were willing to budge on their refusal to raise taxes on America's wealthy, "I do not see a path to a deal."
Republicans said they would not raise taxes and were already taking heat for considering an increase in the legal limit on the debt, which stands at a record $14.3 trillion.
The eight congressional leaders agreed to meet again with Obama at the White House Tuesday afternoon.
Obama said he expected them to meet daily -- including through the weekend, if necessary -- to seal a deficit-reduction deal before Aug. 2. That is when U.S. Treasury officials say the United States will begin to default on its obligations if the debt ceiling is not lifted.
Lawmakers of both parties have refused to raise the debt limit without a plan to restrain the debt in the future.
The real agreement deadline is July 22, to give time for the non-partisan Congressional Budget Office to analyze the deal and for both parties in both chambers to sell the deal to their respective rank and file, officials said.
Monday's meeting focused on a Republican-proposed $2.4 trillion alternative to Obama's so-called $4 trillion grand bargain, officials of both parties said.
The smaller package was under discussion in talks led by Vice President Joe Biden that broke down last month.
In addition to major cuts to domestic agencies, the House GOP proposal calls for slicing $250 billion from Medicare over 10 years, in part by asking well-off seniors to pay more for health coverage, The Washington Post reported.
Obama rejected this, arguing he would not ask "moderate-income seniors to bear $500 or more of additional costs when you couldn't ask the most well-off Americans to give an extra $5 to getting the deficit down," the Post said, citing a Democratic official familiar with the discussions.
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