Senate GOP to Geithner: Default would be your fault

To buttress their argument, Republicans point to figures from the Congressional Budget Office (CBO) as proof the government can cover its debt service payments without borrowing. The CBO states that the government took in $2.23 trillion in tax revenue in fiscal 2011, while only making net interest payments on debt of $213 billion.

They also say a Congressional Research Service report proves that the government need not default on debt payments after hitting the debt limit. Presented to members of Congress earlier this month, it explained that the Treasury has the ability to prioritize spending, and could make sure money went to those debt service payments. CNBC originally reported on the contents of the presentation Wednesday.

Sen. Pat Toomey (R-Pa.) is pushing legislation he says will require the administration to prioritize those payments, which he says removes the threat of a default from the debt-limit debate.

However, Geithner has been unflinching in saying Congress has no choice but to ultimately raise the debt limit. In response to Toomey's bill, Geithner argued that while the government could make payments on its public debt after hitting the debt limit, the world would still see the United States was failing to meet other obligations, which would still be interpreted as a default.

DeMint was joined on the letter by Senate Republican Leader Mitch McConnell (R-Ky.) and Sens. Richard Burr (N.C.), Tom Coburn (Okla.), Thad Cochran (Miss.), Jim DeMint (S.C.) , Orrin Hatch (Utah), Mike Johanns (Neb.), Ron Johnson (Wis.), James Inhofe (Oklah.), Mike Lee (Utah), Rand Paul (Ky.), Rob Portman (Ohio), Jim Risch (Idaho), John Thune (S.D.), Toomey, David Vitter (La.) and Roger Wicker (Miss.).

This post updated at 1:02 pm.