"It's not surprising Republicans are having buyer's remorse, but we need higher revenues now," Fallons said. "The more revenue we raise upfront through a tax rate increase on the wealthy, the less likely the middle class will get hit on the deduction side."
Carney made a similar point, telling reporters that Obama wants to lock in the increased revenue from higher tax rates on top income brackets now. Expected negotiations next year on a broader reform of the tax system would be "revenue neutral," meaning the changes would result in no increase or decrease of overall tax revenue, Carney said.
Conservatives trying to shrink the federal government generally oppose increasing tax revenue. They are particularly opposed to higher tax rates because history shows that once rates go up, it is difficult to later reduce government revenue by lowering them again.
Obama and Democrats argue that increased revenue, including higher tax rates on the wealthy, must be part of broader deficit reduction to prevent the middle class from getting hit too hard.
The president previously said that once Republicans agreed to higher tax rates on wealthy Americans, he would be willing to compromise on spending cuts and entitlement reforms sought by Boehner as part of what the president calls a balanced approach.
Obama's latest offer brought the two sides billions of dollars closer, but it also generated protests from the liberal base of the Democratic Party because it included some benefit cuts in entitlement programs such as Social Security, Medicare and Medicaid.
Justin Ruben, executive director of MoveOn, the liberal movement that backed Obama's presidential campaigns, said the group's members would consider any benefit cuts "a betrayal that sells out working and middle-class families."
In particular, he cited concessions that Obama made in his Monday counteroffer, including a new formula for the consumer price index applied to benefits.
"If such a deal were proposed by the president and speaker, MoveOn members would expect every Senate and House Democrat to do everything in their power to block it," Ruben said. A litany of other progressive groups and politicians also expressed their opposition to what is known as chained CPI.
Carney, however, said Obama's latest proposal includes a provision "that would protect vulnerable communities including the very elderly when it comes to Social Security recipients." He called the president's acceptance of the chained CPI a signal of his willingness to compromise to reach a deal.
The chained CPI includes assumptions on consumer habits with regard to rising prices, such as seeking cheaper alternatives, and would result in smaller benefit increases in future years. Labor unions and advocacy groups for the elderly oppose it.
Over the weekend, Boehner offered for the first time to accept tax-rate increases on household income of $1 million and above, sources said. The speaker also offered to allow the president to raise the debt ceiling in 2013 without a messy political fight, another key Obama demand.
In response, Obama on Monday offered $200 million in new cuts to discretionary federal government spending, divided evenly between defense and nondefense programs.
The president also included for the first time the chained CPI sought by Republicans, and dropped an extension of a payroll tax cut from the past two years.
According to the source who provided CNN with details of Obama's counteroffer, it includes $1.2 trillion in revenue increases and $1.22 trillion in spending reductions.
However, Boehner disputed those figures Tuesday.
"The White House offer yesterday was essentially $1.3 trillion in new revenues for only $850 billion in net spending reductions," he said. "That's not balanced in my opinion."
Obama's latest proposal included $130 billion in spending savings due to changes in the CPI.
However, Republican aides estimate that the CPI would increase tax revenues by an additional $95 billion over Obama's estimated $1.2 trillion, putting the total revenue figure of the president's latest proposal closer to $1.3 trillion.
The GOP aides also do not count Obama's estimated $290 billion in savings on national debt interest accumulation because they do not consider these savings real cuts. Therefore, the Republican aides said, Obama's plan fails to offer a 1-1 ratio of spending cuts and revenue that Boehner would consider balanced.
Carney took exception to the Republican math, telling reporters Tuesday that counting reduced interest on national debt was a legitimate savings counted in budget plans by both parties for years.
Obama and Boehner met face to face at the White House on Monday for 45 minutes, their third such meeting in eight days in a sign of an acceleration in the fiscal cliff negotiations. They also spoke by phone later Monday.
Boehner said he "made it clear to the president that I would put a trillion dollars worth of revenue on the table if he were willing to put a trillion dollars of spending reductions on the table."
Congress had been scheduled to end its work last week, but legislators returned to Washington on Monday and leaders warned members to be prepared to stay until Christmas and return after the holiday until year-end.
Last week, U.S. Rep. Chris Van Hollen, D-Maryland, said a deal would have to be reached by Christmas to allow time for the legislative process to approve the required measure or measures by the end of the year.