In response, Obama and Democrats note that their plan -- already approved by the Senate and needing House approval to be signed into law by the president -- affects just 2% of taxpayers and 3% of small business owners.
While Republicans argue those small business owners account for about half of all business income, Democrats say that's because they include law firms, hedge funds traders and other high-income operations.
Obama and Boehner met face to face on Sunday for the first time since November 16. It also was their first one-on-one meeting in more than a year, when deficit talks broke down.
The outline for a deal has become clear in recent weeks. Both sides agree that more revenue from taxes should be part of the equation, with Obama seeking $1.6 trillion and Republicans offering $800 billion.
A source close to the talks said Tuesday that the White House had floated the idea of dropping the revenue target to $1.2 trillion, then went up to $1.4 trillion on Monday.
Later, Boehner spokesman Michael Steel said his side made a counteroffer to the White House "that would achieve tax and entitlement reform to solve our looming debt crisis and create more American jobs."
Steel also said Boehner and Obama spoke on the phone after the Republican proposal was floated. Steel provided no further details on the counteroffer or the phone conversation, and repeated Boehner's earlier assertion that Republicans were waiting for Obama to identify specific spending cuts he would be willing to make.
Boehner's side wants additional revenue to come from tax reform, such as eliminating some deductions and loopholes, while Obama demands the higher rates on income over $250,000 for families as part of the equation.
Republicans also seek savings from entitlement programs totaling another $800 billion or so, while Obama has proposed $400 billion in reduced entitlement costs. Social Security would not be included in the president's plan.
Another sticky issue -- whether the need to raise the federal debt ceiling early next year should be part of the discussion -- also remains unresolved. Obama says absolutely not, while Boehner says that any increase in government borrowing to pay its bills must be offset by spending cuts.
Graham's comments this week showed that Republicans plan to regroup around negotiations to raise the debt ceiling.
He noted that Obama proposed making permanent a process originated by McConnell that would allow the president to increase the debt ceiling and Congress to then try to block it, an unlikely scenario given Democratic control of the Senate.
"That's going nowhere," Graham told Fox News on Monday, adding: "He's not king. He's president."
However, a top House Democrat said that exploiting the debt ceiling for political gain would backfire on Republicans.
"Threatening to tank the entire economy, which is what would happen if we ever defaulted on our debt, is not a kind of negotiating strategy that is going to be popular with the American people," Rep. Chris Van Hollen told MSNBC on Monday.
It remains unclear if a deal will happen before the end of the year or if the negotiations will carry over into 2013, after the fiscal cliff takes effect.
Without action now on the fiscal cliff, the nonpartisan Tax Policy Center estimates that middle-class families would pay about $2,000 a year more in taxes. Even with a deal, revisions in the tax code and other changes would mean everyone pays a bit more starting next year.
All signs point toward a two-step approach sought by Obama, with initial agreement now on some version of his tax plan with targets set for comprehensive negotiations on a broader deficit reduction deal in the new Congress next year.
Such an outcome would put off the main worry of the fiscal cliff, the expiration of the Bush-era tax cuts that would result in higher rates for everyone.
Obama and Democrats say they would then be ready to negotiate significant savings from entitlement programs, while Republicans say they need to first see commitment on entitlement reforms before accepting any higher tax rates.
McConnell, R-Kentucky, warned on the Senate floor on Tuesday that "until the president gets specific about cuts, nobody should trust Democrats to put a dime in new revenue toward real deficit reduction or to stop their shakedown of the taxpayers at the top 2%."
He and Boehner want Obama to take some of the political heat for proposing cuts to entitlement programs and other government spending.
On Tuesday, congressional Democrats rejected any cuts to the Medicaid health care system for poor and disabled Americans as part of a fiscal cliff deal. The opposition by Democrats showed the pressure Obama faces from his liberal base to avoid significant changes to the entitlement program that benefits millions.
Some in Congress warn that the legislative process will need at least a week to work through potentially complex measures from any proposed deal, meaning there is a de facto deadline of Christmas Day at the very latest for negotiators.
Retiring Sen. Kent Conrad, D-North Dakota, predicted Tuesday on MSNBC that a deal would get worked out in a week's time.