The question now boils down to what President Barack Obama offers in return. Such major questions, still unanswered so close to the end of the year suggest, however, that no spending and tax agreement is imminent.
A source familiar with the Obama-Boehner talks confirmed that Boehner proposed extending low tax rates for everyone who has less than $1 million in net annual income, meaning tax rates would rise on all above that line.
Under current law, the 35 percent top tax rate is scheduled to expire on January 1, and would automatically go to 39.6 percent. Boehner's proposal would allow that rate to rise as scheduled at a threshold of $1 million - putting it back to where it was during the Clinton administration.
The White House has not accepted the proposal and the source could not confirm any additional talks were held on Sunday between Obama and Boehner.
With just over two weeks before the fiscal cliff's $600 billion in automatic tax hikes and spending cuts are triggered, threatening a new recession, there is little time to craft a comprehensive deal that will satisfy both Democrats and Republicans.
Until the latest Republican offer, made on Friday, Boehner had insisted on extending all of the Bush era's lower tax rates, resisting Obama's demand to let the marginal rates rise on income above $250,000. A rising chorus of business executives also had urged Republicans to agree to this.
Some lawmakers and congressional aides had predicted that Republicans, once serious negotiations began, might try to raise the $250,000 threshold, say to $500,000 or $1 million. They also speculated that Republicans, if forced into a tax rate hike on the upper-income groups, might seek a smaller increase, say to around 37 percent.
Although the White House has not accepted Boehner's gambit, it could push negotiations away from entrenched, ideological positions.
"Boehner has now accepted the premise of higher rates. So now we're just arguing over details. I think it's a significant step," said Greg Valliere, chief political strategist at Potomac Research Group.
A framework deal spelling out tax revenue and spending cut targets to be finalized in the new year could be possible, Valliere said.
"Boehner's offer to allow tax rates to go up for taxpayers earning over $1 million fundamentally transforms fiscal cliff negotiations," added Sean West, U.S. policy analyst at Eurasia Group, a political risk consultancy.
In a note to clients, West wrote that it signals, significantly, that Boehner ultimately believes a deal to avoid the cliff is still possible.
"The political burden is now shifted back to the president, who must be willing to take on his party in order to get a deal Boehner can ultimately pass. We do not think the president will overreach: Obama will work with Boehner to get to a deal."
There are still several critical elements to a deal besides a tax rate increase on the wealthy, including Republican demands to cut spending on social programs.
Changes to the expensive Medicare and Medicaid health care programs for the elderly and the poor could be central to any deal, which must also include an increase in the federal debt limit needed by the end of February.
DEMANDS SOCIAL PROGRAM CUTS
Boehner conditioned his tax rate increase offer on Obama's agreement to cuts in social program spending, often called entitlements.
Many Republican lawmakers want to raise the eligibility age for Medicare to 67 from 65. They also want to link Medicare to the income of recipients, making wealthier retirees pay more for their care.
Currently, Medicare does have some means testing, charging higher premiums for coverage of doctors visits and prescription drugs to individuals earning more than $85,000 and married couples earning more than $170,000. Only about 5 percent of recipients pay these higher premiums.
Thus far, Obama has offered only about $400 billion in 10-year entitlement savings, mostly through small adjustments in reining in health care costs - not fundamental changes such as raising the eligibility age.
And just as Boehner faces opposition in his own party to raising any tax rates, Obama faces opposition to cuts to Medicare, Medicaid and Social Security from Democrats, who pledged in election campaigns they would protect these programs.
A major bloc of congressional Democrats has already signaled they will not accept major cutbacks in Medicare as part of any fiscal cliff deal.
House Minority Leader Nancy Pelosi of California and Maryland Representative Chris Van Hollen of Maryland are among the high ranking Democrats in the House who have come out forcefully in recent days against raising the age for eligibility for Medicare to 67 years of age.
"Given the level of savings that is being talked about from Medicare, you can't get it all from providers and drug makers," said Paul Heldman, an analyst at Potomac Research, which tracks Washington policy for investors.
"So opponents of raising the eligibility age have reason to believe beneficiaries will take some sort of hit if a mega-deal is cut," he said.
If Republicans are not successful in securing entitlement program cuts in exchange for a tax-rate increase on the wealthy, they are adamant about using a debt-limit increase as leverage to overhaul Social Security and Medicare.
The U.S. Treasury expects to reach its $16.4 trillion statutory debt cap by year-end, and will exhaust its remaining borrowing capacity around mid-February, risking a potential default.
Louisiana Republican Representative John Fleming, a member of the conservative Tea Party caucus who has never voted to increase the debt ceiling, said he would support a debt limit hike if it were part of a deal to make Medicare and Social Security sustainable.
The pace of activity could pick up the coming week.
House Republicans were told to prepare for a possible weekend session next week, potentially interrupting travel plans for the long Christmas holiday weekend.
House Majority Leader Eric Cantor scheduled "possible legislation related to expiring provisions of law," a reference to the expiring tax cuts, for the end of the week, portending a weekend session. Cantor has said the House would meet through the Christmas holidays and beyond.
(This story was fixed to correct current top tax rate to 35 percent from 36 percent)
(Additional reporting by Thomas Ferraro, Richard Cowan and Kim Dixon; Editing by Fred Barbash, Todd Eastham and Jackie Frank)