Two Democrats voted to move forward with the bill, which stalled in a 53-46 vote.
An article raised questions about Graham's military service.
The Senate will take a vote Monday to move forward with legislation to cut off federal funding.
Manchin is the first Senate Democrat to back the legislation.
"It's time they are investigated," Sen. James Inhofe said.
Nearly every year, a controversy erupts that shakes the political landscape.
Lawmakers kicked the can on a number of political fights until September.
The agency is responsible for "side" agreements with Iran that some lawmakers object to.
The senator is one of nine Republicans who haven't signed on to the defunding legislation.
Presidential candidate says 60-day clock hasn't really even started yet.