Williams, a former Stanford assistant coach and team captain, also interviewed when Jim Harbaugh was hired in 2007.
Yale was 7-3 in 2010 and 11-9 in two seasons under Williams.
The Cardinal were turned down by Boise State coach Chris Petersen, who interviewed last week.
Stanford also interviewed three internal candidates: associate head coach Greg Roman, offensive coordinator David Shaw and defensive coordinator Vic Fangio.
Ivan Maisel is a senior writer for ESPN.com.