What Democrats have consistently said they want is for the House to pass a "clean" spending resolution without any Obamacare provisions, and like one that moved through the Democratic-led Senate.
Democratic Rep. Steve Israel of New York told CNN on Friday that at least 180 of the 200 House Democrats would vote for such a plan.
That means at least 37 Republicans out of the chamber's 233-strong majority would have to defy the party's strategy so far to reach the 217 threshold needed for the measure to pass and go to Obama's desk.
Israel said 20 House Republicans had publicly expressed support for such a move, and that he expected more than enough others to join them if the measure actually came up for a vote.
His fellow House Democrats said Friday they would try to get Republican colleagues to join them in a procedural move that would force a vote on a spending measure with no anti-Obamacare amendments. However, the earliest such a vote could occur under their tactic would be October 14, they said.
"This will at least start the clock ticking," said Democratic Rep. Chris Van Hollen of Maryland.
However, GOP Rep. Tim Huelskamp of Kansas described his caucus as "very unified" and said Democrats are "confused" if they think "we're going to fold and let them win on everything."
Obama and Democrats reject the GOP demands, calling them political extortion intended to force concessions on his signature piece of legislation that was upheld by the Supreme Court last year.
They say they already made a major concession to Republicans when Senate Democrats accepted a lower total funding figure in their proposed spending plan, which would cover the first 11 weeks of the new fiscal year that began Tuesday.
In the view of Democrats, Republicans forced the shutdown and now have no strategy for ending it without getting blamed.
"They're flopping around like dead fish in the bottom of the boat trying to figure out what to do next," Democratic Rep. Jim McDermott of Washington told CNN. "They have no plan B."
House takes piecemeal approach
Boehner and his GOP House leadership, meanwhile, say they will proceed with votes during a rare Saturday session on piecemeal spending legislation to fund popular programs.
House Republican leaders sought to frame the votes as forcing Democrats to go on the record for or against funding for things such as national parks and veterans affairs.
Other piecemeal spending measures would fund the Federal Emergency Management Agency and the National Weather Service, as well as the Head Start program, Majority Leader Eric Cantor of Virginia said Friday.
Democrats oppose the incremental approach, saying it amounts to conservatives choosing to fund programs and services they like.
And Obama would veto such measures if they reached his desk, the White House has said. On Friday, it said the president would sign a measure expected to pass Congress guaranteeing back pay for federal workers who are furloughed because of the shutdown.
The Pentagon may announce as soon as this weekend a plan to bring up to 400,000 furloughed civilian employees back to work, according to two Defense Department officials. CNN has learned the plan is in the final stages of being written and approved.
But even if that happens, another 400,000 government employees could still be furloughed. Others who are considered "essential" would work, albeit they won't get paid until the shutdown ends -- something that could take days, weeks or months more.