The Senate Energy and Natural Resources Committee voted in favor of the legislation Tuesday.
Under the proposal, hunting, fishing and grazing would continue on the 89,000-acre preserve.
The preserve is home to vast grasslands, the remnants of one of North America's few super volcanoes and one of New Mexico's most famous elk herds.
The federal government bought the property from land grant heirs in 2000 with the goal of operating it as a working ranch while developing recreational opportunities for the public.
The transfer was first proposed in 2010 in hopes of getting more consistent funding to preserve as well as improving the area and increasing recreation.