It appeared the Republicans' chances of winning a concession centered on the medical device tax, which was incorporated into the health law to help pay its costs. Some Republicans noted that the Senate has taken a nonbinding 79-20 vote to repeal the levy, and that several members of Obama's party supported the proposal.
In fact, the total was more than half the Democratic senators, 33 in all. Those in favor included members of the leadership, Sens. Dick Durbin of Illinois and Chuck Schumer of New York among them, as well as Murray.
"What will move this is if there are other Democratic senators who put pressure on Reid," said Rep. Richard Hudson, R-N.C.
The 2.3 percent tax, which took effect in January, is imposed on items such as pacemakers and CT scan machines; eyeglasses, contact lenses, hearing aids and other items are exempt. Repealing it would cost the government an estimated $29 billion over the coming decade.
Some Republicans breathed defiance, despite fears inside the party that the GOP would bear the blame for any interruption in government services.
(Read more: Why government shutdowns cost taxpayers big bucks)
"I've got a titanium backbone. Let 'em blame, let `em talk, it's fine," said Rep. Marsha Blackburn, R-Tenn. She said the GOP wanted to keep the government open, but also wanted to reduce its size and "delay, defund, repeal and replace Obamacare," as the health law is called.
Despite such claims, House Republicans have yet to offer a replacement measure for the law they have pledged to repeal. They first promised to do so three years ago, during the 2010 election campaign when they wrested a majority from the Democrats.
If lawmakers miss the approaching deadline, a wide range of federal programs would be affected, from the national parks to the Pentagon.
Some critical services such patrolling the borders, inspecting meat and controlling air traffic would continue. Social Security benefits would be sent and the Medicare and Medicaid health-care programs for the elderly and poor would continue to pay doctors and hospitals.
The new health insurance exchanges would open Tuesday, a development that's lent urgency to the drive to use a normally routine stopgap spending bill to gut implementation of the law.
—By The Associated Press