Paul is highlighting the program as part of "waste report."
The senators pointed to a recent uptick in threats against troops.
The move would make the program available to all veterans currently receiving VA care.
With lawmakers focused on Iran and 2016 looming, its unclear when Congress will act.
The State Department lifted a ban earlier this year.
One group said the would-be Senate Democratic leader's nickname should be "Warmonger Chuck."
Julian Hattem 08/06/15 09:19 PM EDT
The heir apparent to the role of Senate Democratic leader is poised to break with Obama.
The agreement "is not about becoming friends with Iran," Sen. Jeanne Shaheen said.
Her in-state colleague Charles Schumer is perhaps the most important Senate vote on the nuclear bargain.
He said the administration's diplomacy-versus-war argument is "absurd."