The tabloid believe that he has no plans of doing so, and has asked his teammates to clear out his locker on his behalf.
Reports have already suggested that Costa has begun relocating his family to Madrid in readiness for the switch, and now Atleti officials are heading to England in the hope of finally agreeing a fee.
Costa has been omitted from Chelsea's squad for the group stages of the Champions League, which indicates that he is unlikely to play for the club before the January transfer window.
However, Antonio Conte, who told the striker he was no longer wanted at Stamford Bridge, has excluded the Spain global from his Champions League squad.
All other questions relating to Costa were dealt with by Chelsea's head of communications, proving the navigation of the turbulent relationship between the pair will long continue.
The Turkish transfer window closed on Friday evening and Fenerbahce tried to arrange a loan deal for the forward.
He would be able to feature in the Premier League or the Carabao Cup, but even if he were to make an unlikely return to action for the Blues, it's hard to imagine him turning down Atletico when they appear to be just as eager to sign him as he has been to move.
But Costa has said he wants to join Atletico Madrid, who cannot sign him until January because of a transfer ban.