Coaches at some of the most storied programs in college basketball and the sportswear company Adidas AG are at the center of federal allegations that if proven true could shed light on the lucrative worlds of sports management and apparel contracts and underscore the power coaches wield over top recruits.

In one of several alleged schemes outlined Tuesday by federal prosecutors in New York, a top Adidas executive worked with others including a sports agent and a financial adviser to funnel tens of thousands of dollars to the...