This story has been updated.

The House Appropriations Committee is drafting legislation that would allow the government to immediately pay “death gratuity” benefits to families of any troops killed in action during the shutdown.

The bill could be introduced as early as Tuesday, committee spokeswoman Jennifer Hing told Defense One. It’s still being written, but the bill will allow Congress to pay the full $100,000 death gratuity payment that’s supposed to reach the family within 36 hours of a servicemembers’ death.

The House is moving quickly to bring the bill for a vote after public outrage grew when five servicemembers were killed in Afghanistan over the weekend.

"The fact that the House is developing new legislation on this matter shows Congress fully understands that DoD does not have the legal authority to provide these death benefits," a senior defense official told Defense One. Defense Secretary Chuck Hagel "is outraged and disgusted by this situation, but the law as written is inadequate."

And while President Obama said veterans and troops in harm’s way shouldn’t suffer from the shutdown, it’s unclear whether he would sign the bill. When asked during a White House briefing on Tuesday why he hasn’t signed any piecemeal legislation to fully fund essential agencies like the Federal Emergency Management Agency and the Food and Drug Administration, Obama said he’s “tempted.”

“But here's the problem,” he said. “What you've seen are bills that come up where wherever Republicans are feeling political pressure, they put a bill forward. And if there's no political heat, if there's no television story on it, then nothing happens. And if we do some sort of shotgun approach like that, then you'll have some programs that are highly visible get funded and reopened, like national monuments, but things that don't get a lot of attention, like those SBA [Small Business Association] loans, not being funded.”

[Read more at Defense One - Shutdown Shame: Denying Troop Death Benefits]

“And you know, we don't get to select which programs we implement or not. You know, there are a whole bunch of things that the Republicans have said are law that we have to do. And I don't get a chance to go back and say, you know what, this cockamamie idea that this Republican congressman came up with I really don't like, so let's not implement that,” Obama said. “Once you have a budget and a government with a set of functions, you make sure that it's all operating. We don't get to pick and choose based on which party likes what. So you know, that's where the budget discussions take place.”

“Now, if there's some things that the Republicans don't like, they should argue for eliminating those programs in the budget, come up with an agreement with the Democrats. Maybe the Democrats will agree and those things won't be funded. But you don't do a piecemeal approach like that when you're dealing with a government shutdown, OK?” he said.

In the meantime, The Wall Street Journal reported that several charities, including the Lead the Way Fund, have offered to help the families of the five troops killed this weekend. A senior defense official said DoD leaders are "working to connect the grieving families with private organizations that have reached out to us to help while the shutdown continues.

“We are stepping in,” said Lead the Way Fund founder Jim Regan, whose son was killed in Afghanistan in 2007. “We are going to take care of anything that needs to be done.”