And on Wednesday evening, the White House appeared on the verge of a breakthrough, with House Speaker John Boehner (R-Ohio) and Senate Majority Leader Mitch McConnell (R-Ky.) issuing a joint statement pledging to reconsider package swiftly— a signal the two GOP leaders believe they will have the support they need to pull it off.

The administration has spent weeks pushing lawmakers to hand the president fast-track negotiating authority on trade, to guarantee the deal gets a simple up-or-down vote in Congress once it’s finalized. But skeptical House Democrats shocked the administration on Friday by tanking a piece of the package designed to win their support — a measure extending assistance to workers displaced by imports — in order to collapse the whole process.

Now, trade backers think they may have found a way to salvage the initiative. The plan would require seamlessly choreographing and then executing a complex series of maneuvers in both chambers, meaning it comes with a high degree of difficulty. But the idea, while still forming, also extends a glimmer of hope to free-traders where none seemed to exist just a few days ago.

It would go something like this: House Republican leaders would separate the fast-track authority from the worker assistance that Democrats defeated and put it to a vote on its own. GOP leaders would limit defections from the 28 Democrats who helped secure the margin for the fast-track piece of the package on Friday by pledging to bring the worker assistance program up for a vote again later. Meanwhile the Senate, which narrowly passed a measure that combined fast-track with worker assistance last month, would repackage the jobless aid with another, uncontroversial trade-related bill. Then each measure would head to the opposite chamber.

Rep. Gerry Connolly, a pro-trade Virginia Democrat, said Speaker John Boehner (R-Ohio) yesterday gave his cohorts a guarantee that the help for workers — a program known as Trade Adjustment Assistance, which expires at the end of September — would come back up in the chamber, “and he further stated declaratively that it will pass.”

Ultimately, the liberal Democrats who sunk the worker assistance bill will be responsible for reviving it. The conceit behind the new strategy is that once fast-track authority is signed into law, that group will no longer have an incentive effectively to hold it hostage in the hopes of stopping Obama’s broader free-trade push. While Obama once pledged not to sign fast-track authority without the worker help attached, White House Press Secretary Josh Earnest appeared to reformulate the promise on Wednesday, indicating the president would only support a strategy that delivered both to his desk, without specifying that they be linked.

But for the White House, one of the advantages of the new tack is that it end-runs the troublesome House Democrats led by Minority Leader Nancy Pelosi (D-Calif.), whose last-minute decision to speak out against the administration’s plan last Friday helped doom that vote. Instead, the White House is now dealing just with Republican leaders and the fraction of Democrats in either chamber who’ve proved open to supporting Obama’s trade agenda. Administration officials huddled with both groups at the White House on Wednesday, in part to ensure whatever scheme they land on has buy-in from Democrats on both sides of the Capitol.

The strategy, such as it is, appears to be starting in a strong position with the roughly two-dozen business-friendly House Democrats the White House will need. “We take solace in Churchill’s admonition that he’s confident we’ll do the right thing but only after we exhaust all other options first. That’s kind of what we’re doing right now,” Rep. Ron Kind (D-Wisc.), a leader of the pro-trade faction, said after a meeting with his colleagues in the Capitol basement. “The path for us is clear.”

Senate Democrats may be another matter. Only 14 of them joined 48 Republicans to lift the trade package over the upper chamber’s 60-vote threshold last month. That means no more than two can balk at the new plan without sinking it. And many have spent the weeks since the Senate vote enduring high heat from the coalition of labor and environmental groups arrayed against the Pacific trade pact. Then again, the stakes are clear: If the new gambit fails, those Democrats will have taken a tough vote for no reason. Said one senior Senate Democratic aide, “They may just be so desperate to get it done and under so much pressure from the White House to get it done that they just do it.”