House Speaker John Boehner said that the GOP would offer a temporary increase in the debt ceiling in return for discussions with President Obama on other budget and deficit issues.
Republicans have been insistent that budget cuts and other measures be added to the so-called debt ceiling legislation but the aides wouldn't say whether he'll seek to add other material to the measure.
A statement from the White House called Boehner's proposal "an encouraging sign."
Sen. Chuck Schumer (D.-N.Y.) welcomed the news as an indication that the Republicans "are backing off their stand that they have to attach something to it," he told CNBC. "I think they're realizing they were playing with fire."
At the same time he said the short-term nature of the proposal was bad news. "To do this every six weeks would be very devastating to the markets," he said.
The news comes after Treasury Secretary Jacob Lew warned Congress of "irrevocable damage" that an unprecedented federal default could cause, even as House Republicans explored a short-term debt limit increase to provide more time to resolve their budget battle with President Barack Obama.
Lew testified before the Senate Finance Committee on the 10th day of a partial federal shutdown and one week before Lew has said the government will deplete its ability to borrow money. Most economists say the federal default that could result would deal a staggering blow to the world economy, though some Republicans have said the damage would be manageable.
Lew warned that failure to renew the government's ability to borrow money "could be deeply damaging" to financial markets and threaten Americans' jobs and savings. It would also leave the government unsure of when it could make payments ranging from food aid to Medicare reimbursements to doctors, he said.
(Read more: Here are 7 debt-default doomsday scenarios)
"The United States should not be put in a position of making such perilous choices for our economy and our citizens," the secretary said. "There is no way of knowing the irrevocable damage such an approach would have on our economy and financial markets."
As Lew testified, Obama prepared to host top House Republicans at the White House in hopes of finding an opening in an impasse that has shuttered much of the government and threatens federal default. And Thursday morning, a short-term debt limit measure was expected to be a topic at a closed-door House GOP meeting.
It wasn't clear what conditions, if any, GOP leaders might seek to attach to the bill, but conservatives consistently have been pushing Speaker Boehner to add conditions beyond what Obama says he'll accept. The president has said he will negotiate on the budget—but only once Congress unconditionally ends the shutdown and extends the debt limit.
The game of Washington chicken over increasing the debt limit—required so Treasury can borrow more money to pay the government's bills in full and on time—already has sent the stock market south, spiked the interest rate for one-month Treasury bills and prompted Fidelity Investments, the nation's largest manager of money market mutual funds, to sell federal debt that comes due around the time the nation could hit its borrowing limit.
At the Finance committee hearing, Lew met a buzzsaw of incredulity from Republicans, who said the bigger problem was the soaring costs of benefit programs like Social Security and Medicare and the long-term budget deficits the country faces. Many expressed doubt about Lew's description of the consequences of default.
The senior Republican on the panel, Sen. Orrin Hatch of Utah, accused the Obama administration of "an apparent effort to whip up uncertainty in the markets." And veteran Sen. Mike Enzi, R.-Wyoming, said, "I think this is 11th time I've been through this discussion about the sky is falling and the earth will erupt."
(Read more: Shutdown Day 10: Obama seeks opening with GOP)
Lew also rejected GOP suggestions that in the event federal borrowing authority expires, the government could use the dwindling cash it has to make payments to debt holders and other high priority needs. He said federal payment systems are not designed to prioritize and said he didn't believe such an approach was technically possible.
"I think prioritization is just a default by another name," Lew said. He also fended off attempts by the top Republican on the committee, Sen. Orrin Hatch of Utah, and other GOP senators to learn how long a debt limit extension the president would like to see.
"Our view is this economy would benefit from more certainty and less brinksmanship. So the longer the period of time is, the better for the economy," said Lew, who also repeated Obama's willingness to accept a short-term extension for now.
Finance Committee Chairman Max Baucus, D.-Mont., said GOP demands to curb Obama's 2010 health care law as the price for ending the shutdown "is not up for debate" and would not happen.
"We need to reopen the government and pay the nation's bills, no strings attached," said Baucus. Wednesday featured lots of activity but no progress toward ending the budget and debt limit impasses.
Obama had House Democrats over to the White House, while Republican conservatives heard a pitch from the House Budget Committee chairman, Rep. Paul Ryan, R-Wis., on his plan to extend the U.S. borrowing cap for four to six weeks while jump-starting talks on a broader budget deal that could replace cuts to defense and domestic agency budgets with cuts to benefit programs like Medicare and reforms to the loophole-cluttered tax code. Curbs to "Obamacare" were not mentioned.
At the White House, Obama told House Democratic loyalists that he still would prefer a long-term increase in the nation's $16.7 trillion borrowing cap but said he's willing to sign a short-term increase to "give Boehner some time to deal with the tea party wing of his party," said Rep. Peter Welch, D-Vt.
A midday meeting Wednesday between the two top House Republicans and Democrats, meanwhile, yielded no progress.
(Read more: These states are hurt the worst by the shutdown)
Obama also invited the entire House GOP to the White House on Thursday, but Boehner opted to send a smaller squadron of about 20 mostly senior members, which prompted White House Press Secretary Jay Carney to issue an unusual statement criticizing the move to exclude tea party Republicans from the session.
"The president thought it was important to talk directly with the members who forced this economic crisis on the country about how the shutdown and a failure to pay the country's bills could devastate the economy," Carney said.
The frustrating standoff in Washington is weighing down each side's poll numbers, but Republicans are taking the worst drubbing. A Gallup poll put the approval rating for the Republican Party at a record-low 28 percent. Polls have consistently said the Republicans deserve the greater share of blame for the shutdown.0
—By The Associated Press. CNBC contributed to this article.