Senators passed the Bulletproof Vest Grant Program Reauthorization by a voice vote.
The presidential hopeful hasn’t ruled out a vote for the measure.
The GOP leader made a motion Wednesday to move to "fast-track" legislation.
Senators are expected to debate the legislation ahead of a vote on Thursday.
Ted Cruz was one of just three senators to vote against tabling the veto.
Rand Paul and Ted Cruz opposed, while Marco Rubio and Lindsey Graham voted yes.
Corker expects debate on bill will end soon and that deal will be reached.
Rebecca Shabad 05/05/15 12:14 PM EDT
An aide said the 2016 hopeful will be present for the final vote later in the day.
Vicki Needham 05/05/15 11:31 AM EDT
“We have a lot to do, and not much time,” Reid said.
Senators will have 10 hours to debate the House-Senate budget after an initial vote Tuesday morning.