The bill, H.R. 2440, would require Fannie and Freddie to set up a plan for selling "non-mission critical" assets, including patents and historical mortgage data. This requirement is meant to force the GSEs to reduce their footprint in the market as much as possible, which Hurt said last year is an important step because U.S. taxpayers have been backing up the two entities since 2008.

"With more than $150 billion in taxpayer funds already spent propping up Fannie and Freddie and hundreds of billions more a possibility, it is critical that we begin the process of winding down these [GSEs] and increasing the role of the private sector in the secondary mortgage market to strengthen our housing finance system," Hurt said last year when his bill was introduced.

Fannie and Freddie guarantee more than half of U.S. home loans and were put into conservatorship in September 2008, which put them under the control of the Federal Housing Finance Agency. According to the Congressional Research Service, that means taxpayers are standing behind roughly $5 trillion of debt held by the two GSEs.

Many blamed Fannie and Freddie's ability to repackage billions in mortgage loans as mortgage-backed securities as a key factor that led to home-price increases and the resulting housing bubble, but by 2008, it was judged that the failure of either would have led to a broad and severe global financial meltdown.