Updated 3:10 p.m. — Senate Democrats said talks toward resolving the so-called fiscal cliff before the end-of-year deadline had hit a "major setback" on Sunday afternoon due to a standoff over proposed changes to Social Security.

Democrats said that Republicans, led by Minority Leader Mitch McConnell, Ky., are insisting that a deal to resolve the fiscal cliff include what is known as "chained CPI" -- a change in how Social Security benefits are calculated to increase over time.

Just before a self-imposed deadline at which Senate leaders were set to brief their respective caucuses about a prospective deal, negotiations toward a scaled-back agreement to avoid the onset of automatic tax hikes and spending cuts on Jan. 1 appeared on the verge of breakdown.

Related:Obama: GOP's insistence on halting tax hikes for the wealthy is stopping fiscal cliff deal

McConnell said that he had even reached out to Vice President Joe Biden, a former senator who's helped hammer out previous deals, in hopes of jump-starting the talks.

"He and the vice president, I wish them well," Senate Majority Leader Harry Reid, D-Nev., said on the Senate floor.

NBC's Kelly O'Donnell reports Democratic sources say that there has been a major setback in negotiation of a fiscal cliff deal.

"In the meantime, I will try to come up with something," Reid added of Republicans' latest proposal, "but at this stage I don’t have a counter-offer to make."

Obama had offered chained CPI — which would essentially reduce the rate of growth in Social Security benefits over time — as part of a broader "grand bargain" he had previously proposed to Republicans. The GOP rejected that proposal, and moved from there onto House Speaker John Boehner's "Plan B," an ultimately unsuccessful effort.

In his interview earlier today on NBC's "Meet the Press," the president pointed to his offer on chained CPI as evidence of his willingness to compromise in pursuit of a broad fiscal deal.

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"One of the proposals we made was something called Chain CPI, which sounds real technical but basically makes an adjustment in terms of how inflation is calculated on Social Security," Obama said. "Highly unpopular among Democrats. Not something supported by AARP. But in pursuit of strengthening Social Security for the long-term I'm willing to make those decisions."

A Senate Democratic aide said that Democrats had thought such a proposal was off the table, though, as part of the talks toward parried-down agreement.

"It’s basically a poison pill," the aide said of Republicans' demand for chained CPI.

Sen. Lindsey Graham, R-S.C., expressed bewilderment at the breakdown, suggesting that there were more than enough votes for a compromise measure that didn't include chained CPI.

"I don't know what caused this but there's a critical mass of 80 senators who would vote to fix the [alternative minimum tax], the doc fix, extend unemployment insurance, protect everybody 500 thousand and below from a tax increase," he told reporters at the Capitol. "There's 80 senators who will do that without CPI."

McConnell, who spoke briefly on the Senate floor around 2 p.m., struck an ever-so-slightly sunnier note.

"There is no single issue that remains an impossible sticking point," the top Senate Republican said. "I want everyone to know I'm willing to get this done. But I need a dance partner."

Senators are set to huddle with members of their respective parties this afternoon amid votes to discuss the latest as it relates to the fiscal cliff.

As House members return to town this evening for votes this evening, they'll also caucus with fellow party members to discuss what, if any, way forward there is on the fiscal cliff.

NBC's Frank Thorp contributed reporting.