Also, a Washington Post/Pew Research Center survey released Tuesday showed 53% of respondents would blame Republicans for failure to reach a deal, compared with 27% who would blame Obama. A CNN/ORC International poll released last week showed 45% would blame congressional Republicans compared with 34% who would hold Obama responsible.
Despite the public stance softening of some Republicans in each house, signs point to a continuing standoff for now.
Senate Republican leader Mitch McConnell warned Thursday that his party will continue to use the federal debt-ceiling issue as leverage to extract concessions on spending cuts from Democrats, a tactic that Obama opposes.
"The only way we ever cut spending around here is by using the debate over the debt limit to do it," McConnell said in response to Obama's call for removing the issue from political negotiation.
At the White House, spokesman Jay Carney said such brinksmanship over a matter involving a possible unprecedented government default was opposed by the business community and the American public.
"I can't imagine they would want to do that," he said of Republicans.
The House is scheduled to adjourn for the year on December 14, but Boehner's number two, Majority Leader Eric Cantor, said Wednesday the chamber would remain in session until a fiscal cliff deal gets reached.
Boehner, meanwhile, challenged Obama to talk, saying, "I'll be here and I'll be available at any moment to sit down with the president to get serious about solving this problem."
Obama demands that the House immediately pass a measure already approved by the Senate to extend tax cuts from 2001 and 2003 on income up to $250,000 for families.
He contends that both Democrats and Republicans agree that the 98% of American families making less than $250,000 a year should avoid a tax hike when the lower rates from the Bush administration expire on December 31. They call for the House to guarantee that outcome by passing the Senate measure now.
Once that happens, Obama and Democratic leaders promise, they will work out compromises on other spending cuts sought by Republicans to reduce the deficit, such as reforms to the Medicare and Medicaid entitlement programs.
The latest proposal from House Republicans could reduce federal deficits by $2.2 trillion over 10 years, GOP leaders say.
The GOP proposal includes $800 billion from tax reform, $600 billion from Medicare reforms and other health savings and $600 billion in other spending cuts, House Republican leadership aides said. It also pledges $200 billion in savings by revising the consumer price index, a measure of inflation.
While the Republicans gave ground by calling for more revenue through tax reform, the plan mentioned only unspecified elimination of some deductions and loopholes.
Obama's broader deficit reduction plan calls for deficit reduction of about $500 billion over 10 years by limiting itemized tax deductions and other benefits for high-income earners. The 28% limit on deductions would apply to families earning more than $250,000 and individuals making more than $200,000, according to the White House.
Jason Furman, an assistant to Obama on economic policy, told reporters Wednesday that the president wants to ensure additional revenue from higher taxes on the wealthy now to help avoid the fiscal cliff.
"What we're seeking to do is to lock in revenue this year, not have some vague process that may or may not add up to something, you know, in the future, and so we're trying to pass something this month," Furman said.
While the White House has made clear Obama will veto any measure that fails to increase tax rates on the wealthy, aides have signaled a possible willingness to negotiate the specific rate increase.
In an interview with Bloomberg TV this week, Obama said lower tax rates for the wealthy could be negotiated as part of broader tax reform in 2013, but only after those rates increase now.
Obama's deficit-reduction plan would increase taxes by almost $1 trillion over 10 years, a significant portion of a $4 trillion overall deficit-reduction goal.
It also would close loopholes, limit deductions, raise the estate tax rate to 2009 levels and increases tax rates on capital gains and dividends.
The Obama plan includes $50 billion in stimulus spending for programs intended to create jobs, such as repairing roads and bridges.
Experts have said failing to reach a fiscal cliff deal and devise a framework for a broader deficit reduction package to be negotiated when the new Congress is seated in January would cause economic turmoil.
The non-partisan Tax Policy Center estimates that middle-class families would pay about $2,000 a year more in taxes without action.