President Barack Obama on Wednesday suggested that some Republicans had begun to soften their opposition to increased tax rates on the wealthiest Americans, but sharply warned the GOP against looking to tie their budget fight to an upcoming debate over whether to increase the debt ceiling.

Speaking in Washington before a group of corporate leaders, the president suggested some rank-and-file GOP lawmakers might relent in opposing increased income tax rates for the wealthiest earners, meaning Washington is closer to an agreement on the fiscal cliff than principal negotiators' public comments might suggest.

“I think there's a recognition that maybe they can accept some rate increases as long as it's combined with serious entitlement reform and additional spending cuts. And if we can get the leadership on the Republican side to take that framework, to acknowledge that reality, then the numbers actually aren't that far apart,” Obama told members of the Business Roundtable this morning. “Another way of putting this is, we can probably solve this in about a week. It's not that tough. But we need that conceptual breakthrough that says we need to do a balanced plan; that's what's best for the economy; that's what the American people voted for; that's how we're going to get it done.”

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Talks toward resolving the fiscal cliff, the combination of automatic tax hikes and spending cuts set to take effect on Jan. 1, seem to have hit an impasse after House Republicans and Obama each offered countervailing proposals to address the looming combination.

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The stalemate involves the question of whether tax rates are allowed to increase for the top earners. Republicans say they are willing to ask the wealthy to shoulder a larger share of the tax burden, but only through ending deductions and loopholes in the tax code.

"Now the revenues that we are putting on the table are going to come from, guess who? The rich," House Speaker John Boehner, R-Ohio, said Wednesday morning on Capitol Hill. "There are ways to limit deductions, close loopholes and have the same people pay more of their money to the federal government without raising tax rates, which we believe will harm our economy."

Obama, in his appearance shortly following Boehner's press conference, argued that closing those deductions would not raise the type of revenue needed to establish the "balanced" approach on which he campaigned. The president also said that the GOP plan would harm philanthropic and charitable giving.

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As the opposing sides race toward the end-of-December deadline, a New York Times report on Wednesday suggested that Republicans might turn, as a fallback position, to authorizing an extension of current tax rates for all but the wealthiest Americans (this was essentially the president's initial request of lawmakers following the election).

But the report also suggested that Republicans might then use an impending need to authorize more borrowing authority, a point at which the GOP might have more leverage, to extract the kinds of spending cuts and tax and entitlement reforms for which Republican leaders have pushed during this fall's fiscal cliff talks.

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Referencing that report, the president sternly warned Republicans against tying another increase in the debt ceiling to a fiscal fight -- a scenario which would risk reigniting the debt talks during the summer of 2011, during which the government was brought to the brink of default due to partisan differences.

"I have to just tell you, that is a bad strategy for America, it's a bad strategy for our businesses, and it's not a game that I will play," Obama said.

Lawmakers on Capitol Hill are currently set to depart for their holiday recess, though Boehner said that he would remain in Washington, ready "at any moment" to sit down with Obama and negotiate.

Charles Dharapak / AP President Barack Obama pauses as he speaks about the fiscal cliff at the Business Roundtable, an association of chief executive officers, in Washington, Wednesday, Dec. 5, 2012.

But Republicans also clamored for another meeting with the president, expressing their frustration that they had not received any plan in response to theirs from Obama.

"We ask the president, sit down with us, be serious about the specifics and spending so we can stop the wasteful spending in Washington and finally address the problem," said House Majority Leader Eric Cantor, R-Va.