The Spanish giants want to sign Töre before the end of the January as they will be hit with a transfer ban over the summer.
Beşiktaş are reluctant and recently turned down a £7.4m offer from West Ham according to the Sun.
The Black Eagles did however, complete the signing of Armenian winger Aras Özbiliz from Spartak Moscow before loaning him out to Rayo Vallecano.
The arrival of Özbiliz could signal that Töre will be on his way out over the summer but that would be too late for Atleti.
Atleti are reported to have proposed a deal that would see Töre remain at Beşiktaş on loan and move to Spain over the summer.
Töre’s agent Ahmet Bulut recently revealed that the Rojiblancos had also made an offer for the Turkish international winger over the summer transfer window but that the Black Eagles rejected the offer.
“I worked hard to bring Gökhan to Beşiktaş and he is really happy at the club I did offer him to Atletico Madrid over the summer but it did not work out I do not see him leaving over the winter,” Töre was quoted as saying by Fanatik.