The defender has left the Etihad Stadium on a free transfer, but he wants to return in future.
Richards brought an end to his 10-year association with City when he agreed to join Tim Sherwood’s Villa last month, with his contract with the Citizens having come to an end.
He spent last season out on loan at Serie A outfit Fiorentina after City manager Manuel Pellegrini deemed him surplus to requirements at the Etihad Stadium.
Richards admits he is gutted to be leaving, but he insists that he will look to come back in the future, in whatever role he can.
“City have stuck by me and it's definitely the club of my heart,” he told Citytv.
“When I finish I've always said I would want to come back in some capacity. I'm gutted to be leaving but it's the right decision for me and I've got a lot of amazing memories."
Special praise for Roberto Mancini
There were high hopes for Richards when he initially broke into the City first team, with many anticipating him becoming first-choice centre-back for both the Citizens and the England national team for years to come.
Things haven’t worked out that way though, with a series of niggling injuries and the increased competition for playing time at the club severely hampering his development.
Richards was however a regular starter under former manager Roberto Mancini, and he reserved special praise for the Italian in his goodbye interview.
“For me he [Mancini] was amazing," he added. "He played me all the time when I was fit. I probably had my best season under Mancini. He definitely improved me as a player.”