Not only is the answer yes, they would, they have figured out a way to do it, according to National Journal's Congress Daily:
"House Rules Chairwoman Louise Slaughter is prepping to help usher the healthcare overhaul through the House and potentially avoid a direct vote on the Senate overhaul bill, the chairwoman said Tuesday.
"Slaughter is weighing preparing a rule that would consider the Senate bill passed once the House approves a corrections bill that would make changes to the Senate version.
"Slaughter has not taken the plan to Speaker Pelosi as Democrats await CBO scores on the corrections bill. 'Once the CBO gives us the score, we'll spring right on it,' she said."
Each bill that comes before the House for a vote on final passage must be given a rule that determines things like whether the minority would be able to offer amendments to it from the floor.
In the Slaughter Solution, the rule would declare that the House "deems" the Senate version of Obamacare to have been passed by the House. House members would still have to vote on whether to accept the rule, but they would then be able to say they only voted for a rule, not for the bill itself.
SOURCE: Washington Examiner