The former Arsenal director becomes the second high-profile departure from Villa Park this week following former sporting director Hendrik Almstadt who left the club by mutual consent on Tuesday.
"Regarding my resignation, the owner made it very clear last summer that he and I would seek to recruit a Board that would put the Club in a stronger position in the event that a sale didn't happen," said Fox.
"As the Board has taken shape, however, it has become clear that my role also has changed. While I am supportive of where the Board is trying to take the Club, it is my feeling that given the changes they are making, it makes sense for me to seek another challenge."
Villa are currently rock bottom of the Premier League and facing a first relegation in 29 years.
American owner Randy Lerner has announced his intention to sell the club but in financial accounts released last week chief financial officer Robin Russell said "no active discussions were taking place".
Chairman Steve Hollis will become executive chairman while the club searches for Fox's replacement.