Public opinion polls show most Americans unhappy with Obamacare. But surveys also show most Americans opposed to a government shutdown, which would disrupt federal services and deal a blow to the U.S. economy.
Congress must authorize spending in the new fiscal year. The Republican-led House of Representatives last week defied a White House veto threat and passed a bill to keep the government running, but only if Obamacare is defunded.
That bill is now before the Senate, where Democrats that control the chamber vow to remove the provision to defund Obamacare this week and return the measure to the House to sign off on it. It would then by up to House Speaker John Boehner and his Republicans to decide what to do - with time running short.
House Republicans could approve the Senate-passed bill, clearing the way for Obama to sign it into law, or reject it, triggering a shutdown.
(Read more: US House passes bill to kill Obamacare, avert shutdown)
Attaching new conditions
Another option in the House would be for Republicans to attach new conditions to the bill, such as a possible delay in implementation of Obamacare, and send it back to the Senate for its concurrence.
Republican Senator Ted Cruz of Texas, a potential 2016 White House contender, reiterated his support for the effort to defund Obamacare being pushed by the Tea Party movement.
Cruz said he plans to try to raise procedural roadblocks to prevent Democrats from eliminating the defunding provision in the House-passed bill.
Appearing on "Fox News Sunday," Cruz conceded that he does not know if he will have the 41 votes needed in the 100-member Senate to sustain such a roadblock. But he said it's now time for the Senate's 46 Republicans "to unify, to stand together with House Republicans" and against Obamacare.
Republicans need at least 41 votes in the Senate to sustain a procedural hurdle known as a filibuster against the bill.
"Look, this (bill) may end up going back to the House," Cruz said. "I hope and fully believe that the House will continue the fight."
(Read more: House Republicans plan to defund Obamacare: Boehner)
Republican Representative Tom Graves of Georgia said members of his party are "united around a very simple goal, and that is keeping the government open while protecting our constituents from the harmful effects of Obamacare."
"We're going to do everything we can to protect our constituents, and we have eight days to do that," Graves told ABC's "This Week" program.
Democratic Representative Chris Van Hollen of Maryland, also appearing on ABC, said, "There's a reason Republicans senators like Senator McCain and others are saying this is absolutely insane."
"The Republican proposition here is they're going to shut down the government if they can't deny healthcare to millions of Americans, including millions of kids," Van Hollen said.
While a number of lawmakers on both sides of the political aisles said on Sunday talks shows that they do not expect a government shutdown, Democratic House Leader Nancy Pelosi said she believes a lot of conservative Republicans want one.
(Read more: Ripple effect of Obamacare)
"The Republicans put legislation on the (House) floor that was intended to shut down government. That's a victory, because they are anti-government," Pelosi said on CNN's "State of the Union" program.
"I don't paint them all with the same brush, and I certainly don't paint the speaker with that brush," Pelosi said. "But enough of them in their caucus would shut down government."