Washington (CNN) The President's trade agenda scored a major victory Tuesday when the Senate voted to advance a bill to allow "fast-track" approval of large international trade bills.

The outcome of this key procedural vote had been in doubt as a group of 14 pro-trade Democrats weighed whether to continue their support of the bill out of concern that a related workers' assistance package might not pass both chambers.

But after repeated assurances by GOP congressional leaders that workers' assistance measure will be adopted, 13 out of 14 backed the bill.

The vote was 60 to 37, passing by the slimmest margin needed to pass.

A final Senate vote on fast-track could come as soon as later Tuesday, and it will then head to President Barack Obama's desk for his signature.

The vote was especially close as one Republican who supported the measure in the past — Texas Sen. Ted Cruz — said Tuesday morning he would now vote against it.

"... TPA in this Congress has become enmeshed in corrupt Washington backroom deal-making, along with serious concerns that it would open up the potential for sweeping changes in our laws that trade agreements typically do not include," Cruz wrote on Breitbart.

Republican Sen. Mike Enzi of Wyoming, who was absent during the first version of the bill, voted in favor, making up for the loss among GOP senators.

The 14 pro-trade Democrats who supported the first version of the fast-track bill, known as Trade Promotion Authority, when it was packaged with a bill that provides retraining and other assistance to workers who lose their jobs because of large international trade agreements. That bill is called Trade Adjustment Authority.

Passage of the fast-track authority and the workers' assistance bill allows the President to complete a giant Trans-Pacific Partnership trade deal, which would tie the economies of the U.S., Canada and Mexico with several Asian and Pacific nations. Such a deal would also give the U.S. increased influence in the region -- a top priority for the White House.

All the way up until the final vote, it was uncertain whether those Democrats would support the fast-track bill separately from the workers' assistance piece -- as the new legislative strategy calls for -- out of fear stand-alone workers' assistance measure won't get through the GOP-controlled Congress on its own. Trade Adjustment Authority is generally supported by Democrats -- and unions -- and opposed by Republicans. But in a recent legislative tactical move, House Democrats skeptical of fast-track authority recently blocked the trade adjustment portion in hopes of scuttling the fast-track bill.

McConnell went to great lengths Monday to assure reluctant Democrats both bills would get to the President's desk.

"I didn't want anyone to think that we are getting (fast-track) done this week with a promise to get (trade adjustment) done at some later undefined point," he said. "The process this week is clear: We will vote on (fast-track) and then we will vote on (trade adjustment)."

Sen. Ron Wyden of Orego is a co-author of the fast-track legislation, but held back his support until Monday in an effort to ensure trade adjustment portion and two other related bills will get through. He said he was able to convince Republican leaders to add a measure, supported by Democrats and some Republicans, that would boost U.S. steel companies.

"I held round-the-clock discussions with the Senate majority leader, the speaker of the House and leading Democrats over the past week," Wyden said. "I plan to support the continued advancement of the trade package tomorrow."

Another, Sen. Dianne Feinstein, D-California, said she would vote to advance the bill because expanded trade "helps California and the country as a whole."

If the fast-track authority bill reaches final passage, it would go directly to Obama for his signature, because the House already approved it. The Senate would then take up the worker assistance bill, which is attached to a popular Africa trade measure, with a key procedural vote expected Wednesday and final passage by Thursday.

Assuming the trade adjustment measure passes the Senate, the bill would be sent to the House where Democrats would have to consider whether it still makes sense to defeat it. The last time, Democrats were motivated to vote it down in order to block the fast-track authority. This time, that authority will already be on the President's desk -- if everything goes according to plan -- so voting "no" would only hurt the displaced workers Democrats say they want to help.

House Democratic Leader Nancy Pelosi expressed skepticism on Friday that the bill providing assistance and training to displaced workers could move forward under the new legislative strategy.

"I don't see a path for (trade adjustment)," she said.

But other Democratic leaders, such as Rep. Steve Israel of New York, said recently it would be "cutting off our noses to spite our face," if they voted against the Trade Adjustment Authority at this point.

"It's cutting off noses of working people," he said.

If the Senate approves the trade adjustment bill, a vote in the House is expected by Friday.