Sen. Mitch McConnell

Senate Minority Leader Mitch McConnell (R-Ky.) announced Sunday that a Republican offer on the "fiscal cliff" has effectively been rejected by Democrats. (Drew Angerer / Getty Images)

WASHINGTON — Talks over preventing the New Year’s tax hike on all but the wealthiest Americans hit a major setback Sunday when Republicans demanded — and Democrats rejected — a move to trim cost-of-living increases for Social Security recipients.

Negotiations continue, but the offer presented by aides to Republican Senate Minority Leader Mitch McConnell (R-Ky.), was swiftly rejected by Senate Majority Leader Harry Reid (D-Nev.)

It comes as lawmakers convene for a rare Sunday session of Congress, and McConnell said he telephoned Vice President Joe Biden to help broker a compromise.

“I’m willing to get this done, but I need a dance partner,” McConnell. “The consequences of this are too high.”

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Democrats said they viewed the offer as a step backward.

“When Leader Reid received this recent offer he was taken aback and disappointed,” said a Senate Democratic aide granted anonymity to discuss the private talks. “We feel we are further apart than we were 24 hours ago.”

Republicans were already seeking a narrowed definition of wealthy, those earning more than $500,000, and wanted to protect inheritance taxes from a hike, when they tacked on the latest demand to trim benefits.

Adjusting the cost of living for recipients of government benefits, including Social Security, had been offered earlier this month by President Obama in talks with House Speaker John A. Boehner when the two men were negotiating a broader deficit-reduction deal. But once those talks fell apart, the Senate leaders had been trying to cobble together a more modest proposal that would simply avoid the tax hikes coming once the current tax rates expire on Dec. 31.

Reid said he had no immediate counteroffer but might later in the day.

The two sides had been sequestered almost all day Saturday as aides to the top congressional leaders tried to hammer out a compromise.

Obama had initially sought to keep tax rates in place for all but the wealthiest 2% of Americans, those with incomes above $250,000, and once suggested moving to a $400,000 threshold as part of a broader deal that would also include spending reductions and a resolution to the coming debate over the debt ceiling.

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Democrats have been willing to entertain the higher $500,000 income tax threshold Republicans want, but have resisted coupling that with a 35% rate on inheritance taxes for estates valued at more than $5 million. Obama wants the estate tax to rise to 45% on estates above $3.5 million.

With hours remaining to cut a deal, Obama has warned that Democrats will simply call a vote on his initial proposal — preventing tax hikes on all but those incomes above $250,000 – and force the issue. A vote on that could come as soon as this evening if talks break down.

The president’s proposal would also include other must-pass year-end legislation — including an extension of long-term unemployment benefits, which expired Saturday.

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Lisa.Mascaro@latimes.com

Twitter: @LisaMascaroinDC