The Midlands outfit are now searching for a new manager after parting company with former boss Steve Bruce last week following a poor start to the season.
He has been successful at Brentford and had started the season well with them in seventh place while Villa are down in 15th, but only three points back.
Terry made his Chelsea debut as an 17-year-old in 1998 during an EFL Cup tie against the club he would later play for, Aston Villa.
Villa have made an underwhelming start to the Championship season and now sit in 15th position after the first 12 games.
A lifelong Villa supporter, Smith and his family still live in the West Midlands area - both of which are understood to have been significant driving factors in his decision.
The former England captain said he needed some time to mull over his next move after opting to leave Villa after just one season at the club.
Pitarch has previously been director of football at Valencia and Atletico Madrid in his native Spain.
As reported by BT Sport, Grant explains there are only a few clubs around that could be considered a step up from Celtic, hinting that Aston Villa may not be one of them.
"I wish you all the best in this new chapter of your life".
"And Jesus also brings a huge amount of experience in his particular role having worked with some of the biggest clubs in Europe".
Terry joined Chelsea's academy from West Ham at the age of 14 and made his first-team debut as a substitute in a League Cup victory over Villa in October 1998.