Villa, who finished fourth in the Premier League, did not disclose the length of the deal or the fee for the 22-year-old left-back.
British media reported the deal to be worth between £35 and £40 million (US$50.54 million).
Chelsea signed Maatsen from PSV Eindhoven in 2019 and he shone during a series of loan spells at Charlton Athletic, Coventry City, Burnley and Borussia Dortmund, where he spent the second half of last season.
He helped the German side reach the Champions League final against Real Madrid earlier this month having netted in the quarterfinal win over Atletico Madrid.
He is currently with the Netherlands team at Euro 2024 in Germany, where they face Romania in the last 16 on Tuesday.
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