Fast track enables Congress to set negotiating priorities for the administration in exchange for agreement not to amend trade deals. Democrats who fear job losses and Republicans opposed to handing President Barack Obama more authority had hoped to scupper the deal.
"It's a work in progress," Hatch, a Republican, told reporters. Earlier he had indicated that there could be a deal before the end of the day.
Japan, the second-biggest economy in the Trans-Pacific Partnership, has said it sees fast-track as vital to conclude the deal, which would cover 40 percent of the world economy and must pass Congress this year to avoid getting stuck during the 2016 U.S. election campaign.
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Abe is due to address a joint meeting of Congress on April 29 and having fast-track in train by then would be a positive sign.
The bill will restrict Congress to a yes-or-no vote on trade deals in exchange for setting negotiating objectives - giving trading partners confidence deals will not be picked apart - and laying out other ground rules for trade negotiators.
A Democratic aide briefed on the talks said Wyden had won a concession to give lawmakers more oversight on trade negotiations, a key concern for many Democrats.
The aide said the bill would add a new step by requiring the House of Representatives and Senate committees responsible for trade to vote to certify trade agreements meet the objectives set by Congress before going to a yes-or-no floor vote.
Wyden told reporters he wanted to "get trade done right" but gave no indication of the likelihood of a deal.
Unions and trade critics rallied on Capitol Hill on Wednesday against the bill and the vote is still expected to be close as 25 House Republicans have already expressed concern about fast-track, along with 162 Democrats, a total of 187 of 432 current House members.
Hatch said talks were also continuing over extending aid for workers who lost their jobs due to trade. A Republican aide said there were "a number of issues" still in discussion, including details of trade adjustment assistance.