The 24-year-old impressed in the Premier League in 2019/20 despite Norwich's relegation, scoring once and providing seven assists.
This season he has racked up 15 goals and 16 assists in 39 Championship appearances, helping the Canaries back to the top flight.
His stellar performances have been rewarded with a first call-up to the senior Argentina squad for their summer matches.
Villa said in a club statement: "As Emiliano is currently in the Argentina national team’s biosecure bubble, preparing for a World Cup qualifying match with Colombia on Tuesday evening, he will undergo a medical and complete the transfer after the game."
Norwich have announced that the sale of Buendia is a "club-record deal".