Club chairman Randy Lerner has made public his desire to sell the Midlands outfit, though a statement released on Monday declared that no agreement has yet been reached.
On Tuesday, a post on Villa's official website announced that Faulkner was to leave, with chief financial officer Robin Russell stepping in as acting chief executive.
"Having discussed his re-engaging with the chairman's broader business while remaining involved with Villa in a non-operational role, agreement could not be reached," the statement read.
"On the strength of many productive years together, and a longstanding friendship, Paul has parted on good terms."
Faulkner added: "I would like to thank Randy Lerner for the opportunity to become involved with Aston Villa and for the relationship we built over years, which pre-dated my time at the club.
"To the Villa fans, I am grateful for their support over the years, and for their passionate love for and loyalty to this great club, which deserves the very best."
Lerner himself said: "Paul has been fantastic in his role at Villa and more broadly as a colleague over the years. I offer him my deepest thanks."
