While not commenting on specific proposals or counter-proposals, White House press secretary Jay Carney said Obama's requirement that tax rates go up on the top 2 percent of wage-earners "has to be met."
"That remains his position," Carney said during his daily news briefing. "So we have seen since the election a change in tone, in some cases a change in position from different Republicans, including elected Republicans on the issue of, first, revenue, and then acknowledging that rates have to go up."
Carney said Obama's plan so far is the only proposal that achieves balance necessary so senior citizens, the middle class or students aren't unduly burdened.
The country faces a Dec. 31 deadline for lawmakers and the White House to strike a deal to avert a draconian combination of tax hikes and spending cuts, colloquially known as the fiscal cliff.
The meeting came after Boehner proffered a proposal that would call for $1 trillion in new tax revenue -- including increasing tax rates on income greater than $1 million -- and $1 trillion in spending cuts that would include entitlement program reform.
Obama has insisted on higher tax rates on incomes above $250,000 and wants a deal that addresses the $16.4 trillion debt limit.
The White House rejected Boehner's offer, saying it would raise too little money to make a big enough dent in the budget deficit, a Democrat familiar with the talks told The Washington Post. The plan also was rejected because it didn't mention an extension of emergency unemployment benefits.
Carney said the administration "won't tell [Boehner] what his politics are or how he should work with his conference, but we are obviously engaged in conversations with the speaker and his staff and other leadership members, and believe that ... the parameters of an agreement have been and are clear."
Any agreement would not only have to align with Obama's principles, Carney said, "it would require tough choices by both sides, including the president and including Republicans. And that's the only way it can come together."
Obama "believes a deal is possible," Carney said.
Discussions have been "frank and direct and deliberate," Carney said. "And I think we used words like that, and that remains the case. But our goal here is to have an environment that maximizes the possibility of an agreement that works for the American people and that achieves the goals the president has laid out."
"It will not be a deal that any single person or either party is going to say reflects exactly what they wanted," the spokesman said.
The New York Times said both sides expressed new optimism Sunday a deal could be reached this week to avert $500 billion to $600 billion in automatic tax hikes and across-the-board spending cuts due to take effect at the end of the year.
Boehner's offer suggests he expects a big deal with the White House that would produce enough savings to meet his demand that any debt-limit increase be coupled dollar for dollar with spending cuts, the Post said.
The Times said Boehner told Obama Friday he could also accept a deal that would raise $1 trillion in new revenues over 10 years, up from his earlier proposed $800 billion, if Obama committed to significant savings from benefit programs such as Medicare.
Notable deaths of 2014 [PHOTOS]