Earlier Friday, many Republican Senators joined Democrats in voting to end delaying tactics by two conservative lawmakers trying to derail passage of the spending bill.
The 79-19 vote marked an emphatic defeat for Sen. Ted Cruz and Mike Lee, whose tactics have been criticized as futile by many other Republicans.
Obama spoke after the vote from the White House, where he said it was up to House Republicans to follow the Senate's lead and prevent a shutdown. He said the struggle has nothing to do with budget deficits, and said if Republicans "have specific ideas on how to genuinely improve the (health care) law rather than gut it, rather than delay, it rather than repeal it, I am happy to work with them."
He also said even a shutdown would not prevent the scheduled opening of so-called health care exchanges next Tuesday through which millions of Americans will be able to shop for coverage. "That's a done deal," he said
A government closure would force national parks from California to Maine and all the Smithsonian museums in the nation's capital. Workers at the State Department's Bureau of Consular Affairs wouldn't be around to process visa and passport applications, complicating the travel plans of hundreds of thousands.
Washington's bitterly divided lawmakers have managed to come up with last-minute compromises in recent years to avoid shutdowns. Even so, supervisors at government agencies began meetings Thursday to decide which employees would continue to report to work and which would be considered nonessential and told to stay home under contingency plans. Services considered critical to national security, safety and health would continue operating.
No votes on the budget bill were expected in the House until at least the weekend.
An even more complicated and potentially dangerous showdown looms near on the U.S. debt limit. Republican disunity over what to include in that bill has forced leaders to indefinitely delay it. The legislation is required to raise the government's borrowing authority or risk a damaging, first-ever federal default that the Obama administration has warned could occur by Oct. 17.
Many analysts think even the serious threat of a federal default would jar the economy for which neither party would relish being blamed.
On the spending bill, some Republicans want to abandon the effort to kill the health care law and instead insert provisions repealing an unpopular tax on medical devices, or erase federal subsidies for Congress' own health care coverage. They could then dare the Senate to reject the overall measure and face the fallout from the government shutdown that would result.
But lawmakers and Republican aides cautioned that no decisions had been made, in part because it was unclear whether even those provisions would help win enough votes for House passage.
Asked on Thursday whether he envisions the House approving a simple Senate-passed bill keeping the government open, the leader of the House, Speaker John Boehner, told reporters, "I don't see that happening."
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Associated Press writer Alan Fram contributed to this report.
