The situation gives "every appearance of getting dangerously close to the conversation on the debt ceiling," said Nancy Pelosi, the Democratic minority leader of the House of Representatives.
In fact, she said, "We're in the conversation on the debt ceiling."
At the same time, hopes that the debt ceiling fight could be resolved without a catastrophe were raised by reports in The New York Times and Washington Post that House Speaker John Boehner told other lawmakers he would work to avoid default, even if it meant relying on the votes of Democrats, as he did in August 2011.
A spokesman for Boehner would neither confirm nor deny the reports, restating previous public statements by the speaker that "the United States will not default on its debt."
Senator Charles Schumer, the second-ranking Senate Democrat, reacting to the reports, said, "This could be the beginnings of a significant breakthrough."
The New York senator added, "Even coming close to the edge of default is very dangerous," as he urged quick passage of legislation to raise the $16.7 trillion cap on borrowing.
Boehner met behind closed doors with House Republicans on Friday morning.
The Republican-controlled House has so far pressed ahead with a strategy of voting piecemeal to fund publicly popular federal agencies—like the Veterans Administration, the National Park Service, and the National Institutes of Health—that are now partially closed.
(Read more: Washington baits, Wall St. doesn't bite on panic)
Republicans know that neither the Democratic-controlled Senate nor Obama will go along with such an approach, but it allows them to accuse Democrats of working against the interests of veterans, national parks, and cancer patients.
House Republicans on Thursday began lining up 11 more bills to fund targeted programs. These measures are to pay for nutrition programs for low-income women and their children, a program to secure nuclear weapons and non-proliferation, food and drug safety, intelligence gathering, border patrols, American Indian and Alaska Native health and education programs, weather monitoring, Head Start school programs for the poor and other aid for schools that rely heavily on federal assistance.
Disaster assistance also is slated for temporary renewal under the House measures, as well as a bill to provide retroactive pay to federal workers during the government shutdown.
The bills are likely to be debated by the full House over coming days, though not all at once. Democrats have rejected the piecemeal approach and Obama has said he will veto the measures.
The government shutdown and impending debt deadline in the United States kept the dollar near an eight-month low despite signs of a fight back on Friday and drove world shares towards a second week of losses.
The government's September employment report, the most widely watched economic data both on Wall Street and Main Street, had been scheduled for release on Friday, but was a casualty of the shutdown.