WASHINGTON -- In the latest round of the government shutdown showdown, several congressional Republicans are pursuing a policy that they either privately disavow or readily admit is unfair.

In his final offer to the Senate Democrats before the shutdown deadline was hit Monday night, House Speaker John Boehner (R-Ohio) demanded an end to the federal health care subsidies that aides and lawmakers receive when purchasing coverage through the newly created exchanges.

According to Politico, however, Boehner privately worked behind the scenes to save those very same subsidies, which existed before Congress was forced to purchase coverage on the exchanges and are estimated to be worth between $5,000 and $12,000 annually. Boehner's own staffers, moreover, worked diligently to ensure that his negotiations with Democrats to maintain the subsidies remained a secret.

Though many Republican members were willing to vote Monday night for ending the subsidies, at least 20 philosophically oppose the move, a GOP aide on the Hill told The Huffington Post on background in order to speak freely on party divisions.

The move –- which has been championed by Sen. David Vitter (R-La.) -- nevertheless has emerged as the preferred face-saving option for Republicans eager to see an end to the standoff.

After telling reporters that the GOP "can't win" a shutdown fight, Sen. John McCain (R-Ariz.) floated that idea that the Vitter amendment would be a way for House conservatives to potentially claim a modicum of victory.

"I think it would be a very tough vote for [Democratic senators] because overwhelmingly about 97 percent of the American people think that we ought to live like they do, as outrageous as that might seem," McCain said.

But the politics aren't as simple as McCain suggested. In fact, the Arizona Republican himself is torn on the proposal. The Huffington Post spoke with McCain in the halls of the Senate and asked why his and other congressional staffers would have to take a financial hit in order for the government to function.

"I feel terrible for them," McCain said. When asked if his staffers will get screwed by the amendment, he responded, "If it goes through, yeah. I'm already looking at my staff to see how we can adjust the money."

So why pursue a policy that he considers unfair to his own workers?

"It is not a good situation. But when Americans say that they want us to live like everybody else, I understand that," McCain said. "The Office of Personnel Management [OPM] issued a ruling that made us and our staffs in a different category than any other American. And other Americans don’t understand why we would be in a different category."

Others in his party basically agree. Speaking to reporters shortly before the midnight deadline for a shutdown was breached, Sen. Ron Johnson (R-Wis.) also suggested that the Vitter amendment could be a way around the impasse.

"I would like to see just a rejection of the OPM ruling that gives special treatment to Congress," Johnson said. "I think that would be something that would be very difficult for [Senate Majority Leader] Harry Reid to swat away."

Johnson may, in the end, get his wish. House Republicans have shown no sign of backing off the Vitter amendment, at least as of Tuesday morning. But other senators were already looking for alternative routes around the gridlock.

Sen. Rand Paul (R-Ky.) proposed passing a one-week continuing resolution that didn't touch Obamacare for the purposes of letting talks and negotiations continue. He also suggested that he would be fine passing a longer-term continuing resolution to fund the government if it came with some sort of agreement from Reid to pursue health care reforms outside of the context of government shutdown negotiations.

"We could do it that way," Paul said. "If Harry Reid will come forward and say we will fix these problems outside of this deadline and we will fund the government for a month or two, I bet you that will pass."

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    House Speaker John Boehner of Ohio, flanked by Rep. Jeb Hensarling, R-Texas, left, and House Majority Whip Kevin McCarthy of Calif., speaks during a news conference on Capitol Hill in Washington, Thursday, Oct. 10, 2013, following a closed-door GOP meeting, to announce that House Republicans will advance legislation to temporarily extend the government's ability to borrow money to meet its financial obligations. (AP Photo/J. Scott Applewhite)

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  • John Boehner

    Speaker of the House Rep. John Boehner, R-Ohio, speaks during a news conference on the ongoing budget battle outside his office on Capitol Hill on Tuesday, Oct. 8, 2013 in Washington. (AP Photo/ Evan Vucci)

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    Speaker of the House Rep. John Boehner, R-Ohio, pauses during a news conference on the ongoing budget battle outside his office on Capitol Hill on Tuesday, Oct. 8, 2013 in Washington. (AP Photo/ Evan Vucci)

  • John Boehner

    Speaker of the House John Boehner, R-Ohio, makes a statement outside his office to respond to President Barack Obama, Tuesday, Oct. 8, 2013, at the Capitol in Washington. President Barack Obama says he told House Speaker John Boehner he’s willing to negotiate with Republicans on their priorities, but not under the threat of “economic chaos.” (AP Photo/J. Scott Applewhite)

  • John Boehner

    House Speaker John Boehner of Ohio pauses during a news conference on Capitol Hill in Washington, Tuesday, Oct. 8, 2013, as the partial government shutdown enters its second week with no end in sight. Democrats controlling the Senate plan to move quickly toward a vote to allow the government to borrow more money, challenging Republicans to a filibuster showdown as the time remaining to stop a first-ever default on U.S. obligations ticks by. (AP Photo/J. Scott Applewhite)

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    House Speaker John Boehner (R-OH) participates in a news conference after a meeting with House Republicans at the U.S. Capitol, October 8, 2013 in Washington, D.C. (Photo by Mark Wilson/Getty Images)

  • John Boehner

    House Speaker John Boehner (R-OH) speaks during a news conference after a meeting with House Republicans at the U.S. Capitol, October 8, 2013 in Washington, D.C. (Photo by Mark Wilson/Getty Images)

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    House Speaker John Boehner of Ohio arrives for a House Republican conference meeting on Capitol Hill in Washington, Tuesday, Oct. 8, 2013. (AP Photo/ Evan Vucci)

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    House Speaker John Boehner of Ohio prepares leave following a news conference on Capitol Hill in Washington, Tuesday, Oct. 8, 2013, as the partial government shutdown enters its second week with no end in sight. (AP Photo/J. Scott Applewhite)

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    Speaker of the House John Boehner (R-OH) leaves a meeting of the House Republican caucus at the U.S. Capitol October 8, 2013 in Washington, D.C. The U.S. government shutdown is entering its eighth day as the U.S. Senate and House of Representatives remain gridlocked on funding the federal government. (Photo by Win McNamee/Getty Images)

  • John Boehner

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  • John Boehner

    House Speaker John Boehner (R-OH) arrives for work at the U.S. Capitol, October 7, 2013 in Washington, D.C. (Photo by Mark Wilson/Getty Images)

  • John Boehner

    House Speaker John Boehner (R-OH) arrives for work at the U.S. Capitol, October 7, 2013 in Washington, D.C. (Photo by Mark Wilson/Getty Images)

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    House Speaker John Boehner of Ohio arrives on Capitol Hill in Washington, Monday, Oct. 7, 2013. (AP Photo/Susan Walsh)

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    U.S. Speaker of the House John Boehner speaks to the media after a meeting with U.S. President Barack Obama at the White House in Washington, D.C., October 2, 2013, on the second day of the government shutdown. (SAUL LOEB/AFP/Getty Images)

  • John Boehner

    U.S. Speaker of the House Rep. John Boehner (R-OH) speaks to the media after a meeting with U.S. President Barack Obama about the government shutdown on October 2, 2013 in Washington, D.C. (Photo by Win McNamee/Getty Images)

  • John Boehner

    U.S. Speaker of the House John Boehner speaks to the media following a meeting with U.S. President Barack Obama at the White House in Washington, D.C., October 2, 2013, on the second day of the government shutdown. (SAUL LOEB/AFP/Getty Images)

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    U.S. Speaker of the House Rep. John Boehner (R-OH) leaves the White House after a meeting with U.S. President Barack Obama, Senate Majority Leader Harry Reid, Senate Minority Leader Mitch McConnell, and House Minority Leader Nancy Pelosi about the government shutdown on October 2, 2013 in Washington, D.C. (Photo by Win McNamee/Getty Images)

  • John Boehner

    U.S. Speaker of the House John Boehner (C) speaks to the media following a meeting with U.S. President Barack Obama at the White House in Washington, D.C., October 2, 2013, on the second day of the government shutdown. (JEWEL SAMAD/AFP/Getty Images)

  • House Speaker John Boehner of Ohio walks to a House Republican Conference meeting to discuss the ongoing budget fight, Monday, Sept. 30, 2013, on Capitol Hill in Washington. (AP Photo/ Evan Vucci)

  • Speaker of the House John Boehner, R- Ohio, pauses during a news conference after a House Republican Conference meeting about the ongoing budget fight on Capitol Hill on Monday, Sept. 30, 2013 in Washington. (AP Photo/ Evan Vucci)

  • U.S. House Speaker John Boehner speaks to the press at the US Capitol in Washington on October 1, 2013. The White House budget director late September 30, 2013 ordered federal agencies to begin closing down after Congress failed to pass a budget to avert a government shutdown. (NICHOLAS KAMM/AFP/Getty Images)

  • Speaker of the House John Boehner, R-Ohio, with House GOP leaders, speaks briefly to reporters, just after 1am, Tuesday morning, Oct. 1, 2013. (AP Photo/J. Scott Applewhite)

  • U.S. House Speaker John Boehner speaks to the press at the US Capitol in Washington on October 1, 2013. The White House budget director late September 30, 2013 ordered federal agencies to begin closing down after Congress failed to pass a budget to avert a government shutdown. (NICHOLAS KAMM/AFP/Getty Images)

  • Speaker of the House John Boehner, R- Ohio, pauses during a news conference after a House Republican Conference meeting about the ongoing budget fight on Capitol Hill on Monday, Sept. 30, 2013 in Washington. (AP Photo/ Evan Vucci)

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  • FILE - In this Aug. 1, 2013 file photo, House Speaker John Boehner of Ohio speaks during a news conference about budget negotiations on Capitol Hill in Washington. (AP Photo/J. Scott Applewhite, File)

  • FILE - In this Aug. 1, 2013, file photo House Speaker, Republican John Boehner of Ohio, takes questions from reporters at a Capitol Hill news conference on budget negotiations as Congress prepares to leave Washington for a five-week recess. (AP Photo/J. Scott Applewhite, File)

  • FILE - In this Aug. 1, 2013 file photo, House Speaker John Boehner of Ohio speaks on Capitol Hill in Washington about the budget. (AP Photo/J. Scott Applewhite, File)

  • House Speaker John Boehner of Ohio pauses as he takes reporters' questions on NSA leaker Edward Snowden, the economy, and the unfinished work of the House in passing a spending bill, as Congress prepares to leave for a five-week recess, Thursday, Aug. 1, 2013 during a news conference on Capitol Hill in Washington. (AP Photo/J. Scott Applewhite)

  • House Speaker John Boehner of Ohio gestures during a news conference on Capitol Hill in Washington, Thursday, Aug. 1, 2013 that included talk about budget negotiations. (AP Photo/J. Scott Applewhite)