By 9 p.m. Monday, the House had passed two new plans, and a third vote was expected shortly after midnight.
In both the House and Senate, local Democrats voted to leave the health law untouched, joining the rest of their party in rejecting the House approach.
The final House plan would delay by a year the Obamacare mandate that nearly all Americans purchase health coverage. It would also bar the federal government from providing subsidies to help members of Congress, their staff, executive branch appointees, and select others buy insurance.
In Tuesday's early hours, after the deadline had passed, the House planned a vote to reiterate that plan and formally request negotiations with the Senate.
U.S. Rep. Mike Fitzpatrick, a Republican from Bucks County, said the House plan would keep the government running and "makes sure that members of Congress are being held to the same provision of the very same law that they expect the citizens to be held to."
"It's about fundamental fairness," Fitzpatrick said in an interview as the two chambers volleyed GOP demands and Democratic rejections back and forth.
U.S. Rep. Jim Gerlach, a Republican from Chester County, said in a statement that "the House continues to offer sensible proposals to keep the federal government open and ensure fair and equal treatment for all Americans."
"We do not need to close the federal government," he said. "We just need leaders in the Senate majority willing to work as hard as the House has at protecting the American people against some poorly conceived portions of the health-care law."
Obama and Senate Democrats, however, said they would not negotiate with the threat of a shutdown being used as leverage. They said they had long tried to set up formal budget negotiations.
"Keeping the people's government open is not a concession to me . . . it's our basic responsibility," Obama said. "You don't get to extract a ransom for doing your job."
Sen. Robert Menendez (D., N.J.) said Republicans are using the threat of a shutdown to achieve an Obamacare rollback they couldn't win in the 2012 elections or at the Supreme Court. He called those results "democracy in action."
"To put a gun to people's head simply is not acceptable," Menendez said on MSNBC.
Senate Majority Leader Harry Reid (D., Nev.) said, "Tonight we have more proof that House Republicans have lost their minds."
Republicans said they were working to change a law that would hurt the economy and health care.
"There's one egregious flaw after another in the so-called Affordable Care Act that needs serious and honest revision, and there has been a stone wall erected by the Obama administration," said U.S. Rep. Christopher Smith (R., N.J.). "The president has not acted like an adult."
Early in the day, Dent had raised the possibility of rallying moderate House Republicans to support a "clean" spending bill that could pass the House with a mix of GOP and Democratic support and end the standoff.
"I think the overwhelming majority of [House Republicans] does not want the government to shut down," Dent said.
Conservatives, though, wanted changes to Obamacare and Dent could not garner enough votes to alter the House's plans.
If he were going to win over moderates, he would need help from Republicans from the Philadelphia area, in Pennsylvania and South Jersey. They tend to be more moderate than the rest of the GOP House members, have broken with conservatives in the past, and, with narrowly divided districts, stand to lose the most politically from a shutdown.
Democrats attacked Fitzpatrick and others over their votes Monday.
"Fitzpatrick chose the tea party and political brinkmanship over thousands of Pennsylvania jobs when he voted to shut down the government," said the Democrats' congressional campaign arm.
Democrats said Republicans were being guided by their most conservative elements.
"They've got to figure out to what extent they're going to make a rump minority of their majority run the United States government," said U.S. Rep. Robert E. Andrews (D., N.J.).
As the midnight deadline passed, there was no solution in sight. Speeches continued.