“I would hope it gets changed over there,” Rep. Peter T. King (R-N.Y.) told Bloomberg News, echoing other center-right members who explicitly said they were willing to pass the new revision in hopes that the Senate would strip out the harsher provisions.

Even that goal, however, is proving elusive. By late Monday, House leaders had collected more votes than ever but still appeared to be shy of the 216 Republicans they need to pass the measure. They’re stuck between conservatives and moderates, both keenly aware of how they can be attacked on the issue next year.

“If you’re in the House, what you should be thinking now is that if it doesn’t survive, it all comes back to you,” said Sen. Roy Blunt (R-Mo.). “I think what they should be focused on is getting the process moving and, frankly, passing the obligation over to the Senate.”