"One thing I will not compromise over is whether or not Congress should pay the tab for a bill they've already racked up," Obama said in his weekly radio and Internet address.
Democrats said further tax increases for the wealthiest Americans were still possible as Congress looks to close the gap between revenues and expenditures. Democrats point out that Obama has already agreed to significant spending cuts, and that the latest deal only gets the nation to about half of the revenue it needs to resolve the red ink.
"Trust me, there are plenty of things within that tax code — these loopholes where people can park their money in some island offshore and not pay taxes. These are things that need to be closed. We can do that and use the money to reduce the deficit," said Illinois Sen. Dick Durbin, the second-ranking Senate Democrat.
House Democratic leader Nancy Pelosi of California said she, too, wants to put "everything on the table from the standpoint of closing loopholes."
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.