The Iraqis, who worked as interpreters with the military during the war, face violence in their home country, but many of them have been unable to migrate, The New York Times reported.
The original legislation passed in 2007 approved 25,000 visas for Iraqis over five years, but only 8,000 have been issued, documents indicate.
The program has been slowed by security checks, and in Congress, the problem has been stuck in a legislative deadlock, the Times said.
Rep. Adam Kinzinger, R-Ill., said he is working to pass a resolution that would keep the visa program going.
"We're a big country. We're a country with plenty of room. And when you have people willing to fight alongside of us, and put them and their family's lives in danger, then we ought to repay them," he said.