Beto O'Rourke said contractor is asking the Trump Administration to stop sharing information of potential sponsors with immigration agents.
Key Congressional Democrats vow to fight $190 million Trump administration request to run Tornillo.
Both members promise increases oversight of facilities for migrant children when Democrats take control of the House in January.
The private company that has been contracted by the Trump administration is nearing the end of a contract to run the facility.
The refusal by the two firms to take on a yearlong contract came one day before Trump announced the end of family separation.
The move by Sheriff Richard Wiles marks a growing chorus of local law enforcement across Texas that is critical of the federal family separation policy.