Terry, a 40-year-old former Chelsea and England captain,
joined Villa ahead of the 2017/18 season and became the skipper of his new
side.
One of the outstanding English central defenders of his
generation, he played 32 games for the Birmingham club before becoming an
assistant to Villa manager Smith.
Terry, speaking to Villa's website, said: "It has been
a tremendous honour and privilege to have spent these last three years at Aston
Villa, but I feel now is the right time to make the extremely difficult decision
to move on.
"I want to be as respectful to the manager and everyone
at Aston Villa as I can and, having given my future serious consideration over
the summer, I genuinely don't feel it is fair to move into a new season without
being certain of seeing that through."
He added: "My immediate plan is to spend some quality
time with my family and, thereafter, hopefully take up some invitations to
visit clubs and managers around Europe to develop my aim and objective of
becoming a manager.
"It has always been my ambition to move into football
management and, providing the right opportunity presents itself, I feel ready
to take up such a challenge."