Rep. Ron DeSantis's (R-Fla.) bill would not apply to current lawmakers.
The GOP leader made a motion Wednesday to move to "fast-track" legislation.
Martin Matishak 05/06/15 10:45 AM EDT
More than two dozen Republicans are making their case to the Rules Committee.
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.
Rep. Steve King opposes allowing illegal immigrants to serve in the military.
Corker expects debate on bill will end soon and that deal will be reached.
A new House bill would encourage states to pass laws that ban cell phone use while driving.
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.