McConnell has publicly made the case that a default will have serious negative political fallout for the GOP. In an interview on conservative talk show host Laura Ingraham's program on Wednesday, McConnell said Democrats "want to blame the economy on us and the reason default is no better an idea today than when Newt Gingrich tried it in 1995 is that it destroys your brand. It would give the president an opportunity to blame Republicans for a bad economy."

Privately, House Republicans have been driving home the message that a failure to increase the debt ceiling would be a disaster. Sources said Boehner has been "aggressive," in one aide's words, in articulating the need to reach a deal. In a presentation to the House Republican Conference on Friday, Jay Powell, a former Under Secretary at the Treasury Department under George H.W. Bush and a visiting scholar at the Bipartisan Policy Center, laid out just what would happen if a deal isn't reached.

On August 3, according to Powell's presentation, the federal government would be on the hook for $32 billion in committed spending, including $23 billion in Social Security checks, $500 million in federal worker salaries, $1.4 billion owed to Defense Department vendors and $100 million in refunds the IRS owes to businesses. On the same day, the government will take in only $12 billion in revenue, giving the government a $20 billion cash deficit. By August 15, when the federal government is on the hook for a $29 billion interest payment, the cash shortfall will have grown to $74 billion--and possibly more, if interest rates on U.S. debt rises.

The presentation, according to several sources in the room at the time, was aimed at conveying the seriousness of the moment. Recent warnings from Moody's and Standard & Poor's that they could downgrade the nation's debt have underscored the danger a default poses. It's a message Republicans on both sides of Capitol Hill have been trying to drive home to their members.

"There are serious repercussions for the country if the debt limit is not raised and it's important that members have information on what we may be facing. The aim is not to scare anyone but the facts are frightening," said a top Senate Republican aide. But Republican leadership's task is far more complicated than simply convincing their members to vote one way or another. Instead, they must convince those members they are getting a good deal, which, given the slow pace of negotiations, is not a saleable pitch right now. "It would be a mistake to believe that any of our leaders will support a deal that is insufficiently aggressive on debt reduction or one that raises taxes," the House Republican aide said.

McConnell, specifically, has had serious problems with the White House's initial proposals. At one meeting at the White House, McConnell asked how much money a proposal hammered out by Vice President Joe Biden and a bipartisan group of negotiators would cut. Told just $2 billion were enforceable cuts over the next year while the rest of the cuts were subject to ratification by future Congresses, McConnell was incensed. That's when he decided on what one top advisor called his "break glass plan," which would give President Obama the ability to raise the debt ceiling without Republican votes.