In past weeks at secretive White House meetings, Speaker John A. Boehner and President Barack Obama have faced off in fiscal cliff negotiations. But it’s a meeting Tuesday morning at the Capitol that may be Boehner’s toughest negotiating session.

The Ohio Republican is set to meet with his rank and file for the first time since news leaked that he has put tax rate increases and a one-year debt ceiling increase on the table, and one Republican aide predicted “some major pushback” on what would be huge GOP concessions.

But with time running out before the nation goes over the cliff and public opinion turning against them, some Republicans are cautioning not to read too much into any political theater.

“I think it’ll be a faux revolt,” a Republican lobbyist said. “They’ll pretend like they’re upset. But they all know he had to do it. Some of them will truly be upset. But far less than the number of votes they’ll lose.”

A rate increase and hike of the debt ceiling without big corresponding spending cuts runs counter to GOP orthodoxy. Details of the private talks between Boehner and Obama have been closely guarded, though Republican sources indicated they heard more chatter over the speaker’s offer to raise taxes on income more than $1 million and were caught off guard by reports of a one-year offer to extend the debt limit.

But it was unclear whether the White House’s latest offer on Monday evening — tax hikes on those making more than $400,000, a two-year debt limit fix and additional spending cuts —would be received as a step in the GOP’s direction, even as Boehner rejected it.

Still, news of Boehner’s offer provoked strong condemnation from conservative outside groups, offering a hint of where many in the conference are.

“Raising tax rates is anti-growth and raising the debt ceiling is pro-government growth — and this is the Republican position?” said Chris Chocola, president of the Club for Growth.

Rep. Tom Price of Georgia, a doctor and leading conservative, said on Twitter that only significant changes to Medicare “like Premium Support” would qualify as adequate compensation for raising the debt ceiling. Premium support refers to some GOP preferences that Medicare cease to be a fee-for-service program and become one in which seniors are provided a set amount of money to purchase health insurance.

But the reaction in other quarters on Capitol Hill was more subdued, in part because rank-and-file members have been kept largely in the dark by Boehner and his team.

“I don’t know if they’re floating this stuff to see reactions or if it’s legitimately where they’re at,” a second GOP aide said.

A third aide said Republican morale is low. “Republicans know we’re gonna get hammered,” this aide said. “We just want it over with.”

Meanwhile, Senate Majority Leader Harry Reid said he expects the chamber to reconvene on Dec. 26 — the day after Christmas — to wrap its work for the 112th Congress.

“It appears that we’re going to be coming back the day after Christmas to complete work on the fiscal cliff and a few other leftover items,” the Nevada Democrat said Monday.

It was the quintessential Reid-like threat; the majority leader often uses the specter of weekend or holiday work to try to corral colleagues into action. But with so much left to be resolved between the speaker and the president, if their differences can be resolved at all, it seems unlikely that Reid’s words were a bluff.

Though the week between Christmas and New Year’s Day might be spoiled for overworked staffers, the likely session is the strongest indicator yet that top congressional negotiators have hope for agreement. Talks had appeared to be at an impasse as recently as a week ago.

The Senate largely has been in a holding pattern as the House GOP and administration try to forge a deal, and though there is some question about whether there would be enough Republican support for an agreement that raises taxes, many rank-and-file GOP senators have already indicated they’d be willing to break from the party.

“If Boehner can get something passed in the House, I don’t see why it can’t get passed in the Senate,” one top Republican Senate aide said.

With the deadline pressure only intensifying, it’s possible lawmakers who have for the most part been kept out of the loop will be more demonstrative with their frustration over the lack of information or way forward. Both parties have caucus meetings slated for Tuesday afternoon.

At the White House daily briefing Monday, White House Press Secretary Jay Carney was equally opaque about the status of the talks, noting that “conversations continue” and that a plan from Obama is still the “only” one that protects the middle class from bearing too much of the burden of deficit reduction.

Carney ducked questions about whether the president could move off of his position that tax rates must go up for those with an annual income of more than $250,000. There historically has been some division in the Democratic Party on this point, with members such as Sen. Charles E. Schumer, D-N.Y., pushing publicly last year on a $1 million threshold as well.