That's really stretching the definition of private company considering NASA is the major provider of both funding and infrastructure.
You can hardly be a large business in any developed nation without having substantial business relationships with various world governments.
See: Microsoft, Cisco, Oracle, Boeing, Caterpillar, Lockheed Martin, Ford, GM, 3M, Berkshire Hathaway, General Electric, etc.
Might as well pretend Medtronic, Pfizer or Johnson & Johnson aren't private, because they do so much business with government healthcare systems.
Many a time, when companies engage in capital intensive business endeavours they need someone to provide some incentive or even build the initial technology. NASA does this here much like DARPA did with the Internet's initial infrastructure.