A Veterans Affairs watchdog found employees abused their positions for personal gain.
The senators suggested the training program is doing more harm than good in Syria.
He called for a system that "prevents guns from being used by people who should not have them."
The president has pledged to veto the bill.
Sen. Mitch McConnell slammed Democrats, suggesting they flip-flopped on the legislation.
The legislation is spearheaded by Ben Cardin, a Democrat who opposed the Iran deal.
Proposal includes some reductions in mandatory minimum sentencing.
The calls from Senate Democrats are fallout from comments by Kevin McCarthy.
Previous reports say some U.S. troops were told to ignore abuse of young boys by Afghan forces.
The Texas Republican wants to designate Iran's Revolutionary Guard as a foreign terrorist organization.