Republicans have agreed to vote on the bill, H.R. 384, this week, and could approve it as early as Wednesday. Rep. Al GreenAl GreenOvernight Finance: Dems roll out 'Better Deal' economic agenda | Regulators mull changes to 'Volcker Rule' | Gingrich, small biz launch tax cut campaign House Dem seeks to prevent Trump from being able to pardon himself Week ahead: Treasury chief heads before Congress | Senate takes up bank regulator nominees | Big vote set on healthcare MORE (D-Texas), who introduced it in January, says the creation of a special assistant for veterans at HUD would help reduce the more than 22 million homeless vets, and the many more who are at risk of becoming homeless.

"The Homes for Heroes Act of 2013 will improve the coordination of services for homeless veterans and require annual assessments that will help prevent low-income veterans from falling into homelessness," Green said in January.

According to the bill, the special assistant would work to ensure veterans have "fair access to housing and homeless assistance" under HUD programs, and serve as a liaison to the Department of Veterans Affairs. It would also require HUD to issue a report each year on the state of homeless veterans, and the number of veterans being helped by the department.

But Green's bill would also eliminate a position at HUD to create the new veterans' special assistant. Specifically, it would terminate the position of Special Assistant for Veterans Programs in the Office of the Deputy Assistant Secretary for Special Needs in HUD.