The 28-year-old Gaitan, who is under contract at Benfica until June 2019, is currently with the Argentina squad competing at the Copa America. The deal is subject to a medical.
A statement from Atletico read: "The midfielder will travel to Madrid after the tournament that is being held in the United States, to finish all the necessary formalities..."
Gaitan, who has played 252 games and scored 41 goals for Benfica since joining the club from Boca Juniors in 2010, had also been linked with Manchester United in recent months.
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Atletico finished third in La Liga last season and were runners-up to Real Madrid in the Champions League.
(Reporting by Adriana Garcia; Editing by Ken Ferris)
