Majority Leader Mitch McConnellAddison (Mitch) Mitchell McConnellGrassley pressured to run as Democrats set sights on Iowa House Democrats grow frustrated as they feel ignored by Senate Democrats question GOP shift on vaccines MORE (R-Ky.) said Thursday that the Senate could vote early next week on a long-stalled anti-human-trafficking bill.

"It's my hope that we'll be able to go through an orderly amendment process and pass a trafficking bill early next week," he said.

A breakthrough on the Justice for Victims of Trafficking Act appeared Thursday, when a procedural vote was called off to give time for negotiations.

Sen. Debbie StabenowDeborah (Debbie) Ann StabenowHere's evidence the Senate confirmation process is broken Schumer: Democrats considering option to pay for all of infrastructure agenda Democrats closing in on deal to unlock massive infrastructure bill MORE (D-Mich.) said that "at the moment it looks like there's a serious possibility of ... coming to an agreement."

A Republican aide said senators are "closer to a deal than we have been in the past."

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Democrats have blocked the anti-trafficking proposal for weeks because they believe it would expand the Hyde Amendment, which restricts the use of federal funds on abortion.

A way forward on the legislation, however, hasn't been finalized.

"We're not there yet," Senate Minority Leader Harry ReidHarry Mason ReidWarner backing 'small carve-out' on filibuster for voting rights Senate hopefuls embrace nuking filibuster Biden fails to break GOP 'fever' MORE (D-Nev.) said. "I remember we got into a problem with this initially because of the language in the bill, so every word is going to have to be read with this new language ... and then we'll see if we can make it to the finish line."

But he did praise senators on both sides of the aisle for continuing to work toward a solution, adding that "progress has been made."

If senators pass the anti-trafficking legislation early next week, McConnell said that they would then move to Loretta Lynch's nomination to be attorney general, followed by legislation allowing lawmakers to weigh in on an Iran nuclear deal.