Goal had revealed that the Spanish sides have held extensive talks over the France international and, on Monday, a deal was reached with La Liga four weeks away from recommencing.
"Atletico Madrid and Real Sociedad have reached an agreement for the transfer of Antoine Griezmann," a statement on the official Rojiblancos website reads.
"The young French player will undergo a medical before signing a contract with our club."
The 23-year-old came through the ranks of the Sociedad youth academy and had developed into a key player at the Basque side since breaking into the first team in 2009-10.
Griezmann, who was part of the France squad that made it to the quarter-finals at the World Cup in Brazil, has made over 200 official appearances for Real Sociedad.
The gifted attacker had also caught the attention of Arsenal, Tottenham and Monaco, but will continue his career at the Vicente Calderon side instead.
He will become Atletico's fourth summer signing after the previous arrivals of Miguel Moya, Jan Oblak and Mario Mandzukic.