The Arizona Republican is pressing to end what he suggested was a duplicative program.
Steny Hoyer said the freeze made sense in the past, but may limit those who run for office.
The minority leader said passage would "restore the confidence of the American people."
Senators are under a tight schedule to pass legislation ahead of the Memorial Day recess.
Reps. Blum, Graham and Gosar pushed for a vote on the ban.
Sen. Whitehouse's speech marked the 100th in his weekly climate change series.
Monday's votes were the first of approximately 90 amendments that have been tied to the "fast track" trade legislation.
The House approved a package of legislation on Monday to enhance federal assistance programs for veterans.
Rebecca Shabad 05/18/15 05:56 PM EDT
His will would require the Pentagon to disclose total war expenses.
Rebecca Shabad 05/18/15 04:10 PM EDT
A defense contractor in Afghanistan reportedly wasted the taxpayer funds.