I went to a "big 10" school and our athletics budget was crazy (our football coach makes >2 million a year for one thing) but it is all self funded. Tuition does not pay for athletics, and the department actually run a surplus. Though, there are plenty of arguments about if they are solvent enough to weather a bad season, a lot of their budget depends on the sale of football tickets, and they can charge a lot for them as long as the team is up... (Student tickets are $280 a season and depending on the game non-student tickets can go for $200 each plus people "donate" to be able to buy them early).
I've had discussions with sports fans on campus who point out that in addition to funding themselves, the big sports, like football, also make it possible for the non lucrative sports, like track and field or field hockey, to be well funded and offer scholarships.
I'm not sure that it's the case at other schools, but it seems likely that is would be.