House Republicans failed to pin down President Obama on plans to cut spending and rein in deficits during a meeting Wednesday. But at least some came away heartened by what they saw as a new openness by Mr. Obama to potential points of compromise.

In the balance is not only the federal budget for fiscal year 2012, but also a vote to raise the national debt limit before the expected default date of Aug. 2.

In recent weeks, Republicans have been angered by Democrats' repeated attacks against the GOP budget plan – saying that Democrats should then propose their own. In what participants described as the most pointed exchange of meeting, Rep. Paul Ryan (R) of Wisconsin, the architect of the GOP budget plan, called on the president to stop the demagogy over Medicare – a major element of the Ryan budget.

The meeting helped to “clear the air,” said Congressman Ryan. “If we demagogue each other at the leadership level, then we’re never going to take on our debt,” he said at a press briefing after the meeting.

House Republicans are still smarting from the president’s attack on the Ryan plan at an April 13 speech at George Washington University, with Ryan present in the audience at the president’s invitation. On May 24, House Republicans lost what had been a safe GOP seat in New York’s 26th Congressional District after a campaign that focused on GOP's plans to scale back Medicare, making it a subsidy for private insurance.

“We pressed the president pretty hard on a plan – not just an outline – that could be scored by the Congressional Budget Office,” says Rep. Jeff Flake (R) of Arizona, who attended the White House meeting. “Otherwise, he can just hammer us on the Republican plan and not have any plan of his own.”

Others echoed Congressman Flake's sentiment. “We don’t want the Ryan plan to be just used as cannon fodder,” says Rep. John Mica (R) of Florida, who chairs the Transportation and Infrastructure Committee. “We’re trying to get some specificity out of [President Obama]. So far, they haven’t offered that much.”

House Republicans say that they will not agree to support an extension of the national debt limit – currently at $14.3 trillion – unless the White House and the Senate agree to support spending cuts equal to the increase in the debt limit that the president requests.

In response, Obama said that he was willing to take up entitlement reform as part of the discussion. Republicans at the meeting say that the president also said that “everything is on the table” in terms of comprehensive tax reform.

“I hadn’t heard him say that in quite that way,” says Rep. Dave Camp (R) of Michigan, who chairs the tax-writing Ways and Means Committee. “I don’t think you can ever go wrong with a dialogue on these issues, and you don’t get many chances to talk to the president. He was there for more than an hour. That’s important."

But the president also told Republicans that the time had passed for a Democratic plan. Senate Democrats last week announced that they would not be proposing a 2012 budget, as required by law, because ongoing bipartisan negotiations could produce a deal. Obama reiterated that the focus should be on the ongoing bipartisan negotiations, led by Vice President Joe Biden.

House Democrats meet with the president on Thursday to discuss a way forward on debts and deficit. On Tuesday, only 97 Democrats voted to implement President Obama's request to increase the national debt limit without spending cuts. All Republicans and 82 Democrats voted to defeat the measure, which failed 97 to 318.