The young French attacker, who impressed in La Liga last season and at the World Cup, has been lined up by Diego Simeone as a major summer signing.
Negotiations to bring the 23-year-old to the Vicente Calderon began over a month ago but that the two Liga clubs are close to finalizing a deal and confirmation could be imminent.
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Although Griezmann's release clause is €30 million, Real Sociedad is aware it is unlikely to get such an offer from any interested parties and is instead hoping to convince Atletico to part with €25 million.
Atletico's desire to sign the France international, who impressed in his appearances at the World Cup, had largely gone under the radar but they look set to have beaten Arsenal, Tottenham and Monaco to his signature.
Griezmann has been with Real Sociedad since 2005, when he arrived at the youth level, and has a contract at the Anoeta until 2016. However, he has ambitions to play at a club with which he can consistently play Champions League football.
Atletico appealed to the forward as the team showed under Diego Simeone last season - when it won La Liga and was runner up in the Champions League - that it can go toe-to-toe with the best in the country and Europe.
Griezmann would follow striker Mario Mandzukic, signed from Bayern Munich earlier this month, as the latest high profile summer signing from the Rojiblancos, while Manchester United man Javier Hernandez also remains a possible target.