"As President Obama has not put forth a plan that can garner 218 votes in the House, I'd caution him against so hastily dismissing 'Cut, Cap and Balance,'" said Cantor.
Other Republicans, by contrast, took a harder line.
"I find it incredibly ironic that President Obama is one of the few Americans who think we don't need a constitutional amendment 'to do our jobs.'" Said Rep. Jeb Hensarling of Texas, a member of the leadership.
"The point of cutting up the credit cards in order to raise the debt ceiling isn't to meet his tax-and-spend demands; it's to force him to stop spending money we don't have."
Senate Republican leader Mitch McConnell, R-Ky., made a strong statement of support for the measure.
"Not only is this legislation just the kind of thing Washington needs right now, it may be the only option we have if you want to see the debt limit raised at all," he said.
"I strongly urge my Democratic friends to join us in supporting it."
Despite his warning, McConnell and Reid have been deeply involved in writing a fallback measure that is viewed in both houses as promising.
It would allow the president to raise the debt limit by $2.4 trillion in three installments over the next year without a prior vote by lawmakers. Instead, a panel of House and Senate members would be created to recommend cuts in benefit programs, with their work guaranteed a yes-or-no vote in the House or Senate.
Recreating the divide that plagued the earlier negotiations, Democrats want the panel to have the power to recommend higher taxes.
Neither Reid nor McConnell has publicly disclosed the details of the measure, and neither is expected to do so as long as the legislation in the House is pending.
Associated Press writers Andrew Taylor, Erica Werner and Ben Feller contributed to this story
- Running again? Mitt Romney tells Hugh Hewitt...
- University of Phoenix founder dies, leaving...
- 10 things to know about corporate inversions
- A New York Times article said Michael Brown...
- It's about time the government recognize the...
- UN: Ebola cases could eventually reach 20,000
- Is James Foley a martyr? A brutal death...
- Angelina Jolie, Brad Pitt wed privately at...
- A New York Times article said Michael... 41
- Running again? Mitt Romney tells Hugh... 33
- For the first time in American history,... 30
- 10 things to know about corporate... 27
- Doug Robinson: When did Missouri turn... 24
- Obama back at White House after summer... 17
- Why the poverty cycle is harder to... 15
- Winning plaintiffs in 3 states want... 14