WASHINGTON — Narrowing a “fiscal cliff” negotiating gap, President Barack Obama is backing off what had once been ironclad positions.

A new proposal handed to House Speaker John Boehner on Monday drops Obama’s long-held insistence that taxes rise on individuals earning more than $200,000 and families making more than $250,000. He is now offering a new threshold of $400,000 and lowering his 10-year tax revenue goals from the $1.6 trillion he had argued for a few weeks ago. Obama also abandoned his demand for permanent borrowing authority. Instead, he is now asking for a new debt limit that would last two years, putting its renewal beyond the politics of a 2014 midterm election.

And in a move sure to create heartburn among some congressional Democrats, Obama is proposing lower cost-of-living increases for Social Security beneficiaries, employing an inflation index that would have deep consequences, including pushing more people into higher income tax brackets.

Those changes, as well as Obama’s decision not to seek an extension of a temporary payroll tax cut, would force higher tax payments on the middle class, a wide swath of the population that Obama has repeatedly said he wanted to protect from tax increases.

The concessions signal a new stage in the negotiations, where posturing has given way to pragmatism and where both sides still seem willing to lock in on a substantial agreement rather than just putting off a fiscal day of reckoning. To that end, Obama has conceded that a big bargain would require giving up some of his proposals.

“I understand that I don’t expect the Republicans simply to adopt my budget,” he said during his post-election news conference last month. “So, I recognize we’re going to have to compromise.”

The talks, facing a looming deadline, seek to avoid across-the-board tax hikes for nearly all wage-earners as well as spending cuts at the Pentagon and in domestic programs that are set to kick in at the start of the new year. Economists have warned that the combination of the two — the “fiscal cliff” — could stall a weak recovery and threaten a new recession.

Obama’s steps toward Boehner came after the House speaker took a plunge in a call to Obama on Friday — while the nation was focused on the horror of a mass murder in Newtown, Conn. — and agreed to accept an increase in tax rates for taxpayers who earn more than $1 million. Boehner’s plan would raise about $1 trillion in taxes over 10 years.

That was a barrier-breaking moment, changing the negotiations from a fundamental debate over whether tax rates should rise at all to quibbling over who should pay them.

There are still plenty of disputes to iron out. And people familiar with Obama’s proposal were careful not to describe it as his final offer.