House Republicans are anxiously awaiting the release Wednesday of a congressional estimate on their recently passed ObamaCare replacement bill that could force them back to the drawing board.

The GOP-led House passed the bill earlier this month. While President Trump held a celebratory event at the White House to mark the accomplishment, the legislation still has not been sent to the Senate.

That's because the House is waiting for the final audit from the non-partisan Congressional Budget Office which will estimate the bill's impact. And the legislation needs to hit certain deficit-reduction targets -- saving at least $2 billion -- in order to allow the GOP-led Senate to use special rules and pass it with a simple majority.

If the final CBO 'score' finds the House bill does not meet those targets, then the chamber’s GOP leaders may have to rework it and hold another vote.

House Republicans say a re-vote is unlikely. But the possibility has leaders in a wait-and-see mode -- and there's no guarantee a revised bill could pass on a second vote, considering the backlash some Republican members experienced in their districts after the first round.

House Democrats have seized upon the possibility of a do-over, with Maryland Rep. Steny Hoyer, the chamber’s No. 2 Democrat, openly questioning House Majority Leader Kevin McCarthy on the matter.

“Will another vote be necessary?” Hoyer asked the California Republican last week on the House floor.

When McCarthy appeared to answer indirectly, Hoyer continued: “They think they’re not complying with the Byrd rule. … They may have to vote again. … They have no idea what the CBO is going to come up with next week. I would hope the majority leader would give us a clear picture if we are going to consider this bill again or send it to the United States Senate, flawed as it may be.”

The "Byrd rule" is the provision that would allow the Senate to pass such a deficit-reducing bill with a simple, 51-vote majority. This is known as the budget reconciliation process, and could spare GOP leaders from having to corral an almost-impossible 60 votes to pass an ObamaCare replacement bill.

Next year's midterms are looming over the entire process. Even if the House bill is sent to the Senate without a re-vote, the Senate still plans to make changes to the bill -- meaning another House vote could be necessary weeks or month from now.

Even with fellow Republican President Trump touting the legislation from the White House, GOP House leaders had to cancel a vote on the original overhaul bill earlier this year, amid a lack of support. The revised bill passed earlier this month by a 217-213 margin.

Fox News' Joseph Weber, Chad Pergram and Mike Emanuel contributed to this report.