Yet Senate Democrats have dismissed the proposal, and the Obama administration has vowed to veto any such. Democratic Sen. Chuck Schumer of New York called these other GOP offers the latest in a series of "new gimmicks" that avoid the imminent need to fully reopen the government and raise the debt ceiling.
In the Senate, Majority Leader Harry Reid and Sen. Max Baucus filed a proposal Tuesday to raise the debt ceiling without addressing any deficit reduction issues demanded by Republicans. The plan would address the debt ceiling issue through December 31, 2014 -- past the next congressional elections.
Most Republicans would shy away from a bill that doesn't specify spending cuts or other policy changes in return for the increased borrowing authority. Yet Democrats are hopeful some Republicans would vote across the aisle to prevent the potentially catastrophic economic repercussions of a default.
If Senate Republicans require all the time-consuming steps available to them to delay action on the debt ceiling measure, a final vote might not take place until two days before the deadline for raising the borrowing limit, the Democratic aide said.
Debt ceiling 'like the smoke alarm'
Such a Senate measure could increase pressure on the GOP-controlled House to do the same. Yet Republican House leaders have made clear they'll insist on concessions from Democrats before agreeing to raise the $16.7 trillion debt limit.
"The debt ceiling is there for a purpose. It's like the smoke alarm," said House Financial Services Committee Chairman Jeb Hensarling, R-Texas. "Democrats want to unplug the smoke alarm, and Republicans want to go out and fight the fire."
Boehner has insisted a deal to raise the debt ceiling must include deficit reduction steps to lower costs of entitlement programs such as Medicare, Medicaid and Social Security.
However, he and fellow Republicans have moved away from the demand of the tea party conservative wing of his caucus targeting the president's signature Affordable Care Act passed by Democrats in 2010 and upheld by the Supreme Court last year.
Rep. Tom Cole said, while any final deal must include repealing a tax on medical devices that's part of Obamacare, efforts otherwise to tie budget negotiations to the health care law appear dead.
"I think it's been overtaken by the debt ceiling," he said.
On the shutdown, Obama and Democrats say the House would pass a Senate-approved spending plan to end it if Boehner allowed a vote. A CNN survey indicates that -- with 18 Republicans joining all 200 Democrats -- a slim majority in the House would support such a move.
But Boehner won't allow such a vote, Cole said.
"He basically said we're going to have a negotiation," the Oklahoma Republican said.
Poll: Most angry at both parties
While the two parties blame each other in Washington, outside the capital few get off easy.
In a national poll released Monday, most respondents said the government shutdown was causing a crisis or major problems for the country.
The CNN/ORC International survey indicated that slightly more people were angry at Republicans than Democrats or Obama for the shutdown, though both sides took a hit.
According to the poll conducted over the weekend, 63% of respondents said they were angry at the Republicans for the way they have handled the shutdown, while 57% expressed anger at Democrats and 53% at Obama.
"It looks like there is more than enough blame to go around, and both parties are being hurt by the shutdown," said CNN Polling Director Keating Holland.