@ossobuco,
Eons ago (June 8th actually) Tea Party supported Senator Rand Paul (R-KY) proposed an amendment requiring a straight up or down vote in the Senate on increasing the debt ceiling by $2.4Tn (adding to the current ceiling of $14.3Tn - or perhaps raising the ceiling to $14.3Tn - I can't remember which). The House had done the same thing earlier on and the increase was defeated with some Dems supporting all of the Repubs.
Some moderate Repubs supported Paul's amendment and the Dem leadership called it a "gotcha" ploy to embarrass them with the public because of all those 0000000000000000000's.
Sen Jim Demint (R-SC), who still fancies himself as a possible presidential candidate by the way, did threaten to mount a filibuster to require a 60 vote threshold, Thomas.
The amendment disappeared. I suspect that was because Repubs are less sure of who will get the blame if the government defaults (most likely everyone will get the blame).
In his press conference today Obama raised the specter of the government continuing to pay interest on bonds owned by China while stopping food assistance to sick children and checks to retirees. He also noted that between now and August 2nd the House and Senate will only be in D.C. together for a total of 9 days. He seemed to threatened to cut short their holidays.
While talks continue and the clock keeps ticking, there may be some sort of breakthrough on the deficit and the debt ceiling, or all sides may decide to go along with the debt ceiling and move the budget (for the fiscal year starting October 1st, 2011) to become a stand alone issue.
And yes, Cyclo, the House is probably where the fight will take place.