Republican leaders are narrowing in on a strategy to save President Obama's signature trade bill: bring fast-track back up for a vote without the Democratic sweetener of Trade Adjustment Assistance.

A procedural rule vote passed Wednesday, putting Republicans on track to get a clean fast-track, or Trade Promotion Authority, vote as early as Thursday. Although, timing is still in flux.

On Wednesday night, Senate Majority Leader Mitch McConnell and House Speaker John Boehner issued a joint statement, saying, "We are committed to ensuring both TPA and TAA get votes in the House and Senate and are sent to the president for signature. And it is our intent to have a conference on the customs bill and complete that in a timely manner so that the president can sign it into law."

The TAA package was once intended to entice more Democrats to vote for TPA, which limits Congress to a straight up or down vote on any trade deal negotiated by a president over the next six years. But instead of supporting TAA—legislation that aims to give benefits to displaced workers who lost jobs because of trade—unions lobbied Democrats to vote against it in hopes that without TAA, the so-called fast-track bill would sink. In the end, even Democratic Leader Nancy Pelosi, who had been working closely with Republican leaders in negotiations, voted against TAA, and just 40 Democrats supported it.

(RELATED: Three Booted From GOP Whip Team as Leaders Crack Down)

Republican leadership aides say hope has waned since last Friday's vote that the Obama administration will be able to convince dozens of Democrats to flip sides. Republicans believe they have one choice: move ahead with TPA and then try and bring up TAA later.

Now Republicans are working overtime to convince pro-trade Democrats not to walk away from TPA even if it comes without TAA tied directly to it. Both Boehner and Ways and Means Chairman Paul Ryan have been reaching out to Democrats.

Republican leaders have been working behind the scenes to minimize concerns by assuring Democrats that TAA will get its chance on the floor again. The thinking is that if TPA moves forward alone, passes the Senate, and then is signed by the president, union groups will no longer have a reason to lobby against a trade-assistance bill that benefits workers.

"We all want to see this come together, where we have TPA and TAA," Democratic Rep. John Delaney said Wednesday. "If I see that there's a path for that happening that is reasonable and obtainable to me, then I think it's a good strategy and I'll support it." Another House Democrat backing the trade push, Rep. Gregory Meeks, wasn't wavering in his support. "It's too important not to pass TPA and TAA," he said Wednesday.

(RELATED: The GOP's Wish List for Fixing Obamacare)

Even the White House has not completely shut down the strategy.

"It's not clear to us here at the White House that a specific strategy has been settled upon by legislative leaders, who ultimately will be responsible for setting and implementing that strategy," press secretary Josh Earnest said when asked about House Republicans' separating out TAA and TPA. "The one thing that we have been clear about is that the only legislative strategy that the president will support is a strategy that results in both TPA and TAA coming to his desk. ... There are a variety of ways to do that."

Obama held White House meetings Wednesday afternoon with pro-trade Democrats in both chambers "as part of the administration's ongoing outreach on trade," according to a pool report.

After the meeting, Democratic Sen. Chris Coons said, "It was just vigorous discussion, and there is no clear solution in sight yet for what's the best path." Coons, along with fellow Democrat Sen. Tim Kaine, said he needs more assurances than he has right now that they'll be able to get TAA along with TPA.

(RELATED: Here's What Top Defense Officials Told Congress About Their ISIS Strategy)

White House officials met with pro-trade Democrats Tuesday to discuss the best way to move trade. Then, in a rare showing of bipartisan cooperation between House Republicans and the White House, Boehner followed up with many of the same Democrats on Capitol Hill in hopes of building trust and shoring up their votes on TPA.

The heavy lift will be that even if Republicans can pass TPA out of the House without TAA attached to it, Senate Democrats remain skeptical of that approach. The two bills were interlocked intentionally in the Senate in order to win Democratic support. When asked earlier in the week if the Senate could pass TPA without TAA, Democrat Maria Cantwell simply said, "No, no, no."

This story has been updated with more reporting from the Hill.


Alex Brown and Alex Rogers contributed to this article

We want to hear what you think about this article. Submit a letter to the editor or write to letters@theatlantic.com.