President Barack Obama has agreed to curtail future cost-of-living hikes for Social Security and softened his demand for higher taxes on the rich in quickening talks with House Speaker John Boehner to avoid the ‘‘fiscal cliff,’’ various media reported Monday night, citing people familiar with the talks.

Both sides expressed hope a deal was near, The New York Times reported.

Speaking a few hours after Obama and Boehner met at the White House, sources told the Associated Press the president was now seeking a higher tax rate beginning at incomes over $400,000, up from the levels of $200,000 for individuals and $250,000 for couples that were cornerstones of his campaign. That would mean only the top rate, 35 percent, would rise to 39.6 percent. Boehner has offered to raise rates starting at $1 million.

Reuters said its source stressed that the Obama offer was not final.

Talking Points Memo reported that the White House terms also include new revenue of $1.2 trillion rather than an initial $1.6 trillion, $1.2 trillion in spending cuts, a "fast track process" for corporate and individual tax reform in the new Congress, extending unemployment benefits set to expire at the end of the year, additional stimulus in the form of infrastructure spending and a two-year debt limit increase.

Boehner's latest proposal calls for $1 trillion in new tax revenue, which would come from raising rates and limiting deductions that the wealthiest can take.

The White House proposal would also allow the cut in the payroll tax, which funds Social Security, to expire and cut $130 billion from benefits by adjusting the program's inflation index. That is a clear concession to the Republicans, but offer would include protections for the poorest recipients, a source said.

Obama’s offer does not include raising the age of Medicare eligibility from 65 to 67, a Republican goal that has drawn particularly strong resistance from Democratic liberals.

Several officials also said Boehner’s offer late last week to accept higher tax rates included provisions that would mean higher taxes on investment income and dividends earned by wealthy Americans and also in the estate tax.

The people who described the talks did so on condition of anonymity, citing the secretive nature of the discussions.

Obama’s plan would also secure some tax and policy changes long sought by both parties but unattainable in the context of smaller budget deals, the Times reported. It would permanently extend popular business tax breaks like the credit for corporate research and development, permanently stop the expansion of the alternative minimum tax so it does not affect more of the middle class, and stop a long-planned and deep cut to Medicare health providers that Congress has never allowed to take effect.

To avoid more fiscal showdowns, Obama wants the federal borrowing limit to rise high enough to take the issue off the table for two years, although he said Congress could periodically try to override a presidential lifting of the debt ceiling. That would be hard to stomach for Republicans.

Obama’s offer also would set the top tax rates on dividends and capital gains at 20 percent, Businessweek reported. Combined with tax increases from the 2010 health care law scheduled to take effect in January, the top rates would be 23.8 percent.

The president would return the estate tax to 2009 rules, with a $3.5 million per-person exemption and a 45 percent top rate.

Boehner spokesman Brendan Buck said, "Any movement away from the unrealistic offers the president has made previously is a step in the right direction, but a proposal that includes $1.3 trillion in revenue for only $930 billion in spending cuts cannot be considered balanced."

After the new Obama position came out, the liberal group swiftly threatened to mobilize its members against it, taking particular issue with cuts to entitlements.

"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," said Executive Director Justin Ruben in a statement.

TPM reports, however, that some Democrats and liberal think tanks would prefer tying Social Security benefits to the lower “chained CPI” over raising eligibility ages, if a choice must be made.

Senate Majority Leader Harry Reid, D-Nev., said his chamber will wrap up work on the issue after Christmas, Reuters reported.

"It appears that we're going to be coming back the day after Christmas to complete work on the fiscal cliff," he said on the Senate floor.

Boehner faces a crucial test Tuesday morning when he is expected to brief his caucus in the House. He is not expected to bring any deal up for a vote unless a majority of the 241 House Republicans support it.