Richards, 26, has agreed a four-year contract and will move to Villa Park on July 1.
The right back helped City win the Premier League title in 2012 but rarely played for the first team under current manager Manuel Pellegrini and was loaned to Italian Serie A side Fiorentina last season.
"I'm sure he will prove to be an excellent signing for the football club," Villa manager Tim Sherwood told the club's website.
Richards has played 13 times for England and made nearly 250 appearances for City.
(Reporting by Ed Osmond, editing by Pritha Sarkar)
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