The White House and Congress continued their awkward tango near the edge of the fiscal cliff Tuesday, publicly lamenting long-held policy disagreements while House Republicans and President Barack Obama privately exchanged plans.

Late Tuesday afternoon, the office of Speaker John A. Boehner, R-Ohio, divulged that Republicans had made another counteroffer to the White House, in response to a previously unreported second offer from Obama. In that offer, the president reduced his demand for new revenues from $1.6 trillion to $1.4 trillion.

Yet neither the speaker nor GOP aides would outline the specifics of their latest proposal, beyond stating that the GOP plan “would achieve tax and entitlement reform.” An administration official confirmed that the White House presented House Republicans with a new proposal Monday, adding that the president and speaker spoke by phone Tuesday evening after the White House received the GOP’s response.

In a sign of how closely held negotiations are, even GOP leaders leaving a Tuesday meeting in Boehner’s office were unaware any new offers had been exchanged. Upon exiting, incoming GOP Policy Chairman James Lankford, R-Okla., said he believed the talks were at a stalemate, but he later clarified that leaders did not discuss the latest House GOP nor White House offer at the meeting.

“I was unaware that we had already sent something over,” Lankford said. He noted that the House has “one negotiator,” in Boehner.

But whether the offers being passed back and forth will lead to an eventual deal remained a question. “It’s more serious than the press release kabuki theater of two weeks ago, but less serious than it needs to be to actually get a deal,” said one Congressional aide.

The news of a new GOP plan came after all four top Congressional leaders issued public remarks that varied little from the talking points they have been following for weeks as they each accused the other of stalling or not being serious.

Boehner said in the morning he was “optimistic” about the prospects of averting scheduled tax increases and unwanted automated spending cuts, but he accused Obama of slow walking the process. Senate Majority Leader Harry Reid, D-Nev., hours later said it would be “extremely difficult” for a deal to get done before Christmas.

Republicans are asking the White House to specify additional spending cuts, although it’s unclear how much of the scheduled $1.2 trillion in across-the-board discretionary spending cuts beginning Jan. 1 are acceptable to the GOP as part of that demand. And Democrats are continuing to ask Republicans to agree to increased revenues, especially higher rates on highest earners.

“As the Speaker said today, we’re still waiting for the White House to identify what spending cuts the president is willing to make as part of the ‘balanced approach’ he promised the American people,” Boehner spokesman Michael Steel said in a statement announcing the latest GOP offer.

But perhaps the most important development Tuesday was the subtle way some GOP leaders hinted that they may be looking to gain leverage over the White House by caving on tax rates now. Though GOP members state their opposition to tax hikes, they pointed to a growing consensus within the party that the president has leverage over them because of November’s election results and the tax rates’ Dec. 31 expiration. The thinking is that if Republicans wave the white flag on taxes now, they could take up the armory again in 2013 when the government hits its debt limit.

Senate Minority Leader Mitch McConnell, R-Ky., seemed to indicate he believed this, too.

“What the president’s trying to achieve on the top two tax rates, you know, he can get by doing nothing. The law is certainly stacked in his favor,” he told reporters.

“Tax reform, obviously, can’t be done between now and the end of the year. I believe we have a bipartisan view that — that after 25 years, it’s time for tax reform again, but that’s going to take a while,” McConnell said, before adding of a larger debt deal, “We now have another opportunity here at the end of the year to try to engage that discussion again. We’ll have another opportunity later when the debt ceiling issue arrives. … You’d think this whole discussion was about nothing other than raising the top two tax rates.”

On the House Republican side, top members were skeptical about the progress of talks with the White House.

“I think the president keeps changing what he’s asking for, so [I feel] less confident,” Majority Whip Kevin McCarthy, R-Calif., told reporters, when asked whether he is more or less confident.

It could take time to ink out the legislative language of any agreement, not to mention the 72 hours the House GOP’s rules dictate bills must be posted on the Internet for public review.

Reid had a similar message, but a different scapegoat.

“Until we hear something from Republicans, there’s nothing to draft,” the Nevada Democrat said.

And White House Press Secretary Jay Carney rebuffed Boehner’s repeated calls Tuesday for Obama to offer up specific new cuts to entitlement programs, suggesting that the GOP should make specific suggestions first.

But Carney perhaps summed up what every party at the negotiating table has been attempting to do. After earlier saying he would “neither confirm nor deny” progress, or lack thereof, in the talks, Carney said, “I’m trying to be incredibly opaque.”