The maneuver would let them approve the bill without actually voting on it. CBS News correspondent Sharly Attkisson explains how that can happen.
If Democrats get the votes, here's how it would work:
The House votes on the health care bill that the Senate passed December 24 -- the one with those controversial sweetheart deals for select states, like the, the Louisiana Purchase, and what some see as federal funding for abortion. Next, the House would vote on an updated version that fixes things they don't like, like those sweetheart deals. That becomes a "reconciliation bill." The Senate passes that, and it's a done deal.
The problem is, Democrats don't yet have the votes to do it that way. So House Speaker Nancy Pelosi is considering a controversial strategy to help give cover to those vulnerable Democrats facing tough elections.
Under the House "self-executing rule," members vote only on the reconciliation bill, the one with the fixes. The controversial Senate version is automatically "deemed" to be "passed" - deem and pass - without a separate vote.
Republicans say it's unconstitutional, and have taken to calling the approach the "Slaughter rule," named for Louise Slaughter, the top Democrat on the House Rules Committee overseeing the process.
"If they are going to slaughter democracy, they shouldn't do it in public," said Greg Walden, R-Ore.
The namesake herself begs to differ.
"There's no way in the world we would be doing an unconstitutional thing," said Slaughter, D-N.Y.
Both parties use the self-executing rule. It's how they passed the smoking ban on domestic flights, an employment verification measure, and a census bill.
Some legal scholars say the method raises serious questions.
"The constitution says both houses, the House and the Senate, must pass an identical bill which then is passed on to the president for his signature before it would become law," said CBS News chief legal correspondent Jan Crawford. "That doesn't look like it would be the case here."
So why hasn't the self-executing rule gotten much attention before? It's never been used on anything this big. Democrats know it could add to the appearance that backroom deals are pushing through something a lot of Americans don't like, but it may be the only way the White House believes it can get done.