Senate Majority Leader Harry Reid of Nevada said the Democratic-led chamber will not relent.
"The Senate will never pass a bill that guts the Affordable Care Act," Reid declared.
A partial government shutdown would keep hundreds of thousands of federal workers off the job, close national parks and generate damaging headlines for whichever side the public held responsible.
Washington faces two deadlines: The Oct. 1 start of the new budget year and a mid-October date now estimated for the 17th when the government can no longer borrow money to pay its bills on time and in full.
The first deadline requires Congress to pass a spending bill to allow agencies to stay open. The mid-month deadline requires Congress to increase the government's $16.7 trillion borrowing cap to avoid a first-ever default on its payments, which include interest obligations, pension benefits, payments to thousands of contractors large and small, and salaries for the military.
The standoff just four days before the end of the fiscal year increased the possibility of a shutdown, with no signs of compromise.
The No. 2 Democrat in the Senate, Dick Durbin of Illinois, said that because of the time it takes the Senate to approve even non-controversial bills, if the House amends a Senate-passed spending bill and returns it to the Senate over the weekend, "That is a concession on their part that we're going to shut down the government."
Not far from the Capitol, at a community college in Largo, Maryland, Obama insisted he would not negotiate over his signature domestic achievement, either on a bill to keep the government operating or legislation to raise the nation's borrowing authority.
"The entire world looks to us to make sure that the world economy is stable. You don't mess with that," Obama said of the debt ceiling/default measure. "And that's why I will not negotiate on anything when it comes to the full faith and credit of the United States of America."
Republicans who lost the presidential election and a shot at Senate control last year are trying to use must-pass measures to advance agenda items that the Democratic-led Senate and Obama have soundly rejected. The last-ditch effort on "Obamacare" comes just days before coast-to-coast enrollment in the plan's health care exchanges begins Oct. 1.
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Associated Press writers Andrew Taylor and Donna Cassata contributed to this report.