Shutdown Day 10: Short-term debt ceiling proposal on lawmakers' minds House Republicans considering plan to temporarily raise debt ceiling GOP leaders will meet with President Obama on Thursday afternoon Obama signaled willingness to consider short-term extension Wednesday Washington (CNN) -- It's not a total solution, and there's still plenty of work to be done, but House Republican leaders are considering a plan to temporarily raise the nation's borrowing limit while keeping the partial government shutdown in place, members and leadership sources told CNN. The proposal would include demands for negotiations on issues related to the national debt, the sources said, although it's unclear precisely what form those talks would take. It's also unclear how long an extension of the debt ceiling would last, although senior House Republican sources have previously indicated an extension of four to six weeks seems likely. House Republicans are scheduled to meet Thursday morning to discuss the idea, while GOP leaders will travel to the White House in the afternoon to meet with President Barack Obama. Senate Democrats have a separate meeting scheduled with the president. At a private White House meeting with House Democrats on Wednesday, Obama said he would consider a short-term deal to raise the federal borrowing limit, a Democratic lawmaker told CNN. "If that's what (House Speaker John) Boehner needs to climb out of the tree that he's stuck in, then that's something we should look at," according to the lawmaker, who attended the meeting and spoke on the condition of anonymity. The revelation comes amid a stalemate that has Republicans trying to use spending and debt limit deadlines as leverage to wring concessions from Obama and Democrats. The result of the political standoff has been a partial government shutdown that is in its second week and fears of a possible U.S. default on its debt that economists warn could cause another recession. Measure of hope The lawmaker left the White House meeting with a "small measure of hope that this becomes an exit strategy." The president warned that if Republicans want to propose a short-term fix and Democrats say no, then the firm ground Democrats think they're standing on now would soften. Boehner and Republicans are demanding that Obama and Senate Democrats negotiate on deficit reduction steps that would be part of legislation to reopen the government and raise the limit on federal borrowing needed to pay the bills. Obama has refused to enter formal talks until the shutdown ends and the debt ceiling has been raised to remove the threat of default. A senior House Republican told CNN that GOP members may be willing to go for a short-term debt ceiling hike -- lasting four to six weeks -- as long as the president agrees that negotiations will occur during that time. But Republican sources have also told CNN the parameters of those talks have to be specific enough to sell to their skeptical GOP members. So what would Democrats be willing to talk about? The lawmaker said House Democrats are open to hearing what Republicans have in mind with a short-term deal. Paul Ryan steps into budget fight GOP seeks leverage The shutdown began when Congress failed to pass a spending plan for the new fiscal year that started October 1. Now another deadline looms: the need to increase the federal borrowing limit by October 17 or risk a U.S. default. Days of back-and-forth rhetoric and jibes between the leaders have brought no direct negotiations, but plenty of accusations and political spin. On Wednesday, GOP leaders appeared to shift their focus from efforts to dismantle Obama's signature health care reform, the initial driving force behind the shutdown, to securing spending cuts elsewhere. Boehner, who earlier this year told his GOP colleagues that he was finished negotiating one on one with the president, has called for Obama to sit down for what he calls a "conversation" on how to reopen the government and prevent what would be the first-ever U.S. default as soon as next week. But when Obama invited the entire House Republican caucus to the White House as part of a series of meetings with legislators, Boehner's office responded that only the GOP leadership and committee chairmen would attend the Thursday gathering. "It is our hope that this will be a constructive meeting and that the president finally recognizes Americans expect their leaders to be able to sit down and resolve their differences," said a statement by a Boehner aide. Obama's invitation was intended to demonstrate outreach to Republicans just eight days from when the Treasury says Congress must increase the federal debt ceiling or risk default. White House spokesman Jay Carney said Obama was disappointed that Boehner was limiting attendance at Thursday's meeting to fewer than 20 of the more than 200 House Republicans. "The president thought it was important to talk directly with the members who forced this economic crisis on the country" about the potential harmful effects of the shutdown and a possible default, Carney said in a statement, repeating that Obama "will not pay the Republicans ransom for doing their job." Meanwhile, GOP leaders were distancing themselves from demands by tea party conservatives to also make dismantling Obamacare a condition for agreement. Supporters stand by representatives and government shutdown "Right now, we need to find common ground," he wrote in the column posted online Tuesday night. Perhaps in response to a conservative backlash, Boehner made a brief statement Wednesday on the House floor that focused on the GOP message that Obamacare was detrimental to the country. He stopped short of linking it to any negotiations on ending the shutdown and raising the debt ceiling. Boehner insists that the government must reduce deficits, declaring that Republicans won't raise the debt ceiling without steps toward that goal. But a House GOP leadership source told CNN on Wednesday that Obama's rejection of linking negotiations to raising the borrowing limit meant that Republicans probably would be forced to agree to a "clean" debt ceiling limit proposal in exchange for setting up talks on deficit reduction steps. According to the source, the economic implications of a U.S. default "scares people" to make such a deal acceptable to enough House Republicans in order to get negotiations started. The source acknowledged that Boehner may lack support from some or most of his GOP caucus, requiring Democratic votes for the proposal to pass. Democrats planning Democratic Sen. Dick Durbin of Illinois told CNN that his side already was mulling over "what we will discuss, what we will negotiate over, what things will be on the table." The GOP-led House passed a measure Tuesday to set up a special negotiating team comprising members of both parties from the House and Senate, but Obama and Democrats rejected the concept as the latest Republican gimmick to force talks before raising the debt ceiling. The business community has called for resolving the Washington stalemate to avoid a default that would spike interest rates and affect the economy. According to the poll conducted over the weekend, 63% of respondents said they were angry at Republicans for the way they have handled the shutdown, while 57% expressed anger at Democrats and 53% at Obama. According to a new Gallup poll Wednesday, the Republican Party's favorable rating dropped to 28%, down 10 percentage points from September. The 43% for Democrats was a 4-percentage-point drop from last month. Senate Democrats sought to keep up the pressure, holding a rally on the Capitol steps Wednesday to demonstrate their unified stance in calling for House Republicans to reopen the government and raise the debt ceiling. A CNN survey indicates that enough Republicans in the House would join Democrats in voting for a Senate-passed spending plan to end the shutdown. All 200 Democrats and 19 Republicans support passing a continuing resolution with no additional legislative strings attached. With two vacancies in 435-member House, 217 votes are the minimum needed for the measure to win approval in the House. Doing so now could cost him his leadership post due to the conservative backlash it would probably unleash, analysts say. CNN's Chelsea J. Carter, Paul Steinhauser, Jim Acosta, Deirdre Walsh, Barbara Starr, Ted Barrett, Dan Merica and Brianna Keilar contributed to this report.