WASHINGTON—Divisive trade legislation survived a key vote in the Senate on Thursday, bringing the measure one step closer to what's expected to be a more difficult path in the House next month.

The so-called fast track bill, which President Barack Obama wants in order to finish up and enact a sweeping Pacific trade agreement, cleared a Senate procedural vote 62-38, a narrower margin than last week, when a procedural move to advance the bill won 65 votes.

Republicans widely support fast track, which could pave the way this year for a congressional vote—without amendments or procedural delays—on the Trans-Pacific Partnership agreement with Japan and 10 other nations.

Thursday's vote is the latest barometer of fast track's popularity, which faces wavering support among a handful of key Democrats, and the first key test since Sen. Maria Cantwell (D., Wash.) sought to tie the measure's advancement to the reauthorization of the Export-Import Bank.

This week Ms. Cantwell, who backed the bill in committee, said she wouldn't vote to move it further through the Senate unless Republican leaders advance legislation to renew the bank, which provides financing and credit guarantees to exporters, led by Boeing Co., which has a manufacturing base in Washington state.

On Thursday morning Ms. Cantwell, along with Democratic Sens. Patty Murray, of Washington, and Heidi Heitkamp of North Dakota met with Senate GOP leaders. Ms. Cantwell said they were "trying" to strike a deal over the Export-Import Bank, but hadn't gotten any assurances both chambers would vote on reauthorizing the agency before it expires at the end of June.

"We just want a vote in both the House and the Senate before the bank expires," Ms. Cantwell said as she left Senate Majority Leader Mitch McConnell's office.

Democrats said Thursday that Mr. McConnell had agreed to hold a vote in June over the Export-Import Bank.

Supporters of the fast track bill will still need to steer the measure past other procedural hurdles as well.

Last week Sen. Ron Wyden (D., Ore.), one of the architects of fast track and related legislation, led a group of protrade Democrats who temporarily withdrew their support of the bill, also known as trade promotion authority, until Republican leaders agreed to move a bill with measures to boost trade enforcement and expedite customs.

The Obama administration and Mr. McConnell (R., Ky.) have sought to expedite debate and passage of fast track so that the House can consider the bill and so that the Pacific deal has a chance of getting completed and passed before the peak of the presidential election season, which can be fatal to trade deals. Mr. McConnell is also seeking to move other legislation this week, and debate could extend into the weekend.

But Democrats opposed to the legislation, led by Sen. Sherrod Brown of Ohio are warning that Mr. McConnell isn't allowing enough amendments and debate on the legislation. "We're fast tracking this whole idea of a fast track process," Mr. Brown said Thursday. "Why is this good for our communities?"

In the Senate, the bill still faces additional procedural hurdles, long stretches of time to debate the bill and potentially a handful of amendment votes before a final vote on the trade bill could occur. Though senators often agree to speed up the process, it's not clear whether trade opponents will agree to do so.

Write to William Mauldin at william.mauldin@wsj.com and Kristina Peterson at kristina.peterson@wsj.com

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