WASHINGTON — The government limped into a third day of partial shutdown Thursday with no sign of a way out after a White House conversation between President Barack Obama and top congressional leaders seemed only to harden the stances of Democrats and Republicans.

The dearth of progress deepened worries about a bigger problem rumbling ever closer — a mid-October deadline for raising the government’s borrowing limit before it runs out of money to pay creditors. The U.S. Treasury warned on Thursday that failure to raise that debt ceiling could spark a new recession even worse than the one Americans are still recovering from.

“The president remains hopeful that common sense will prevail,” the White House said in a written statement after the unproductive meeting about the political standoff that has idled 800,000 federal workers and halted an array of services Americans expect from their government.

House Speaker John Boehner, R-Ohio, complained to reporters that Obama used the meeting simply to declare anew that he won’t negotiate over his health care law.

House Republicans, pushed by a core of tea party conservatives, are insisting that Obama accept changes to the health care law he pushed through three years ago as part of the price for reopening all of government. Obama refuses to consider any deal linking the health care law to routine legislation needed to extend government funding or to raise the nation’s debt limit.

“We’re probably through negotiating with ourselves,” Rep. Jeb Hensarling, R-Texas, chairman of the House Financial Services Committee, said Thursday on MSNBC.

Republicans who initially sought to defund the health care law in exchange for funding the rest of government have gradually scaled back their demands but say they need some sort of offer from Obama.

Expressing frustration after Tuesday night’s White House meeting, Boehner said: “All we’re asking for here is a discussion and fairness for the American people under Obamacare.”

The White House said Obama would be happy to talk about health care — but only after Congress moves to reopen the government “and stop the harm this shutdown is causing to the economy and families across the country.”

If the shutdown dispute persists it could become entangled with the even more consequential battle over the debt limit. The Obama administration has said Congress must renew the government’s authority to borrow money by Oct. 17 or risk a first-ever federal default, which many economists say would dangerously jangle the world economy.

Treasury’s report Thursday said defaulting on the nation’s debts could cause the nation’s credit markets to freeze, the value of the dollar to plummet and U.S. interest rates to skyrocket.