Vietto joined Villarreal last summer from Argentinian side Racing Club on a reported 5.5million euros transfer deal and signed a five-year deal with the Yellow Submarine.
"We have sold Vietto to Atletico Madrid," Villarreal supremo Roig said in a press conference on Monday.
Vietto enjoyed an impressive debut season in the Primera Division scoring 12 goals in 32 league appearances to help the club finish sixth in La Liga.
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Villarreal have reportedly accepted a £15million offer from Atletico to acquire the 21-year-old striker.
Atletico coach Diego Simeone knows Vietto well having signed the player from Estudiantes while he was in charge of Argentinian side Racing Club in 2011.
Simeone's side are keen to strengthen their attacking line this summer after selling Croatian forward Mario Mandzukic to Juventus on Monday.
Atletico finished third in the Primera Division last month, and will play in next season's Champions League.
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