Meanwhile, GOP leaders were distancing themselves from demands by tea party conservatives to also make dismantling Obamacare a condition for agreement.
Ryan's plan drops Obamacare
Republican Rep. Paul Ryan of Wisconsin, the House Budget Committee chairman who was the party's vice presidential nominee last year, argued in a Wall Street Journal op-ed that Democrats and Republicans should focus on "modest reforms to entitlement programs and the tax code."
"Right now, we need to find common ground," Ryan wrote in the column posted online Tuesday night. "We need to open the federal government. We need to pay our bills today -- and make sure we can pay our bills tomorrow. So let's negotiate an agreement to make modest reforms to entitlement programs and the tax code."
However, Ryan's column never mentioned Obamacare, focusing instead on forced spending cuts to domestic and military programs, as well as reforms to Medicare.
Ryan's Obamacare omission appeared to anger conservatives, who took to Twitter in response.
"Much like White House press, Paul Ryan doesn't mention Obamacare in WSJ oped," tweeted Dan Holler, spokesman for the conservative group Heritage Action.
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 Republicans would likely 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 Boehner may lack support from some or most of his GOP caucus, requiring Democratic votes for the proposal to pass.
Democratic Sen. Dick Durbin of Illinois told CNN his side already was mulling over "what we will discuss, what we will negotiate over, what things will be on the table." Obama also met with the House Democratic caucus at the White House.
The GOP-led House passed a measure on 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.
Meanwhile, Senate Democrats announced they will propose a measure to increase the debt ceiling beyond next year's congressional elections with no additional issues attached.
While many Republicans are certain to oppose it, Democratic leaders hope increased pressure for Congress to prevent a default next week will cause some GOP senators to vote for it.
A GOP source told CNN on Tuesday that the White House was having corporate chief executives call Republican leaders. The business community has called for resolving the Washington stalemate to avoid a default that would spike interest rates to impact the economy.
Without a breakthrough, the shutdown would continue at a cost estimated at up to $50 billion a month. Failure to raise the debt ceiling by next week's deadline would leave the government unable to borrow money to pay its bills for the first time in its history.
All the partisan bickering -- and lack of progress -- is taking its toll not just on furloughed workers, shuttered government facilities and programs, but also on Americans' confidence in their government.
Poll: Most angry at both parties
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.