The Huffington Post’s Jennifer Bendery reports that 17 House Republicans have now come out in support of a “clean CR” that would temporarily fund the government at sequester levels, without defunding or delaying Obamacare. Presuming all House Dems would also vote for it, that means a majority of the House now supports keeping the government open.

Of course, the House GOP leadership won’t allow any vote on such a measure, because it would pass and the government would stay open without Dems making unilateral concessions. But is there anything Congressional Dems can do to procedurally prod House Republicans along, or to somehow capitalize on 17 House Republicans who are ready to fund the government?

There’s been some talk about a “discharge petition,” but Sarah Binder explains why that’s a nonstarter.

However, according to Congressional scholar Norman Ornstein, there are a few options available that might not get the result Dems want, but would still increase the hardball quotient:

1) Republicans are expected to vote today on piecemeal funding of the government, including a measure funding the National Institutes of Health. Senate Dems will spike this measure, and House Republicans will accuse them of refusing to help children with cancer, as they already have done today.

Ornstein says Senate Dems could take that NIH measure, and amend it to restore funding to original pre-sequester levels, and send that back to the House. Republicans would then have to vote for higher funding (angering conservatives) or allow Dems to accuse them of refusing to fund the NIH. “Amend it to fund NIH at a robust level for a full year, send that back to the House, and make them explain why they don’t want to fund research grants,” Ornstein says. Remember, sequester cuts to NIH are already taking a toll.

2) House Republicans are set to vote on other piecemeal measures, such as bills funding national parks, veterans programs, and payments to the National Guard. Ornstein says Senate Dems could theoretically take one of those and attach the Senate Dems’ “clean CR” funding the whole government at sequester levels, and challenge Republicans to vote against that.

This would be a stunt, and no vote would happen, but it would be a good way to call the bluff of House Republicans who say they want Dems to “negotiate” over them to find a way out of the impasse. “All of this becomes a sham in a sense,” Ornstein says. “But I could easily see Democrats amending one of these to incorporate the Senate budget that was passed. Send Republicans the Senate budget and say, ‘Okay, you said you want to go to conference. Let’s do it.'”

Of course, House Republicans would never allow a vote on anything like this, since (again) it doesn’t extract unilateral concessions from Dems in the form of undermining Obamacare. Indeed, it looks borderline impossible to force Republican leaders to allow a vote on keeping the government open without defunding or delaying Obamacare, even though it now looks like a majority of the House of Representatives supports this course of action.