But McConnell bluntly declared that the "tax issue is over" after last week's agreement.
"We don't have this problem because we tax too little; we have it because we spend too much," McConnell said.
Making the rounds on the Sunday talk shows, McConnell was asked repeatedly whether Republicans were prepared to see the nation default on its spending obligations. McConnell said that wouldn't be necessary, so long as Obama agrees to the spending cuts.
But at one point, when asked by NBC's David Gregory whether the GOP strategy will be to hold the debt ceiling "ransom" in exchange for spending cuts, McConnell said it was a "shame we have to use whatever leverage we have" to get the president's attention.
"None of us like using situations like the sequester (automatic across-the-board spending cuts) or the debt ceiling or the operation of government to try to engage the president to deal with this," McConnell said.
Sen. Lindsay Graham, R-S.C., didn't dismiss the idea of allowing a partial shutdown of government until an agreement can be reached. Texas Sen. John Cornyn and other Republicans have floated the idea of a shutdown as a way of winning deeper spending cuts.
"I believe we need to raise the debt ceiling, but if we don't raise it without a plan to get out of debt, all of us should be fired," Graham said.
Maryland Rep. Chris Van Hollen, the top Democrat on the House Budget Committee, said the Republican strategy amounted to: "Give us what we want ... or we're going to tank the United States economy."
Pelosi said she believes the president has enough authority under the 14th Amendment to raise the debt ceiling without Congress' blessing. But the White House has said previously that it does not believe that the amendment — which says the "validity" of public debt shouldn't be questioned — gives the president that power.
McConnell spoke on NBC's "Meet the Press," ABC "This Week" and CBS "Face the Nation." Pelosi was on CBS. Durbin and Graham appeared on CNN's "State of the Union" and Van Hollen was interviewed on "Fox News Sunday."