House Republican leaders unveiled a plan Tuesday to keep the government open past Sept. 30, but were scrambling to build support within their own ranks after conservatives savaged the proposal for failing to defund President Obama’s health initiative.

The plan, as presented to the party’s rank and file in a closed-door meeting Tuesday morning, calls for the government to be funded at current levels through Dec. 15, continuing the sharp budget cuts known as the sequester.

It would also include a resolution to defund the Affordable Care Act, commonly known as Obamacare. But the Democrat-controlled Senate could vote that down and send the underlying budget bill on to the White House for Obama’s signature, avoiding a government shutdown when the new fiscal year begins Oct. 1.

House Speaker John A. Boehner (R-Ohio) and Majority Leader Eric Cantor (R-Va.) told reporters the strategy was aimed at keeping the government open while forcing Senate Democrats to stage a vote on the health law, which is hugely unpopular among some voters.

“We will send to the Senate the provision which says, up or down, are you for defunding Obamacare or not?” Cantor said. “The House has taken a stand numerous times on its opinion of Obamacare. It’s time for the Senate to stand up and tell their constituents where they stand on this atrocity of a law.”

Estimates on when the government might hit the debt limit.

Tea party activists and conservative political groups were unconvinced. For weeks, they have pressed Boehner to shut down the government or refuse to raise the debt limit unless Obama agrees to undermine his signature legislative achievement. On Tuesday, a flurry of angry e-mails from the Club for Growth and the Senate Conservatives Fund flooded Capitol Hill inboxes, threatening to punish lawmakers who support Boehner’s gambit.

“First Boehner betrayed us by supporting Obama’s war in Syria. Now he wants to trick you into supporting ObamaCare,” said an e-mail to supporters from Matt Kibbe, president of FreedomWorks, a political group once run by Boehner’s friend, former House majority leader Richard K. Armey. “Boehner is jumping through hoops to help Barack Obama and betray the American people.”

Meanwhile, Sen. Ted Cruz (R-Tex.) blasted the plan at a tea party rally on Capitol Hill. “If House Republicans go along with this strategy,” he said, “they will be complicit in the disaster that is Obamacare.”

The noisy protests were hitting the mark with some lawmakers. “There’s going to be tremendous pressure on the conference to vote against this idea,” said Rep. John Fleming (R-La.), who said he would not support the budget bill because Boehner and Cantor refused to commit to push for a one-year delay of Obamacare as part of the forthcoming fight over the federal debt limit.

Others said they were inclined to support the budget strategy, arguing that Republicans have little hope of winning the battle to stop Obamacare. “By and large, people understand Obamacare is not our creation,” said Rep. Tim Griffin (R-Ark.). “We’re doing the best we can to stop this, and this may be our best option.”

House Minority Whip Steny H. Hoyer (D-Md.) said Democrats would not support the budget bill. Aside from the provision regarding Obamacare, Hoyer said, the plan to continue the sequester is “totally unacceptable and irresponsible.”

In the Senate, however, Democrats are likely to accept the House bill and move on to the next big fight over the federal debt limit, a senior Democratic aide said.

Congressional leaders are scheduled to meet Thursday to discuss that battle. According to an independent analysis of projected cash flow released Tuesday, the Treasury will run short of cash to pay the nation’s bills as soon as Oct. 18 unless Congress agrees to raise the debt limit.

While the Treasury may be able to stretch available funds into early November, lawmakers would risk causing chaos in world financial markets if they waited that long to act, the report by the Bipartisan Policy Center found.

“What’s being suggested here is that you better be acting by Oct. 18, or you’re going to have problems,” said G. William Hoagland, the center’s senior vice president and a former budget adviser to a long line of Senate Republican leaders. “We’re playing with matches.”

The nation hit the debt ceiling on May 19. Treasury Secretary Jack Lew has been employing a variety of emergency borrowing measures since then, but by the end of August had only about $108 billion at his disposal, the report said. When that money is gone, the Treasury will be forced to rely entirely upon incoming revenues and will slowly run short of funds to pay all its bills.

The danger zone extends from Oct. 18 to Nov. 5, with big payments due on Oct. 23 and especially Nov. 1, when Treasury must make $58 billion in payments to Social Security recipients, Medicare providers and active-duty military.