The Brazilian-born Spain international has been strongly linked with the Blues of late, with Jose Mourinho rumoured to have made him his top target for the summer transfer window.
Various sources have suggested that a deal which could rise to around £50m is all but sealed, with the striker set to pen a contract worth around £185,000-per-week.
But Atletico manager Diego Simeone has met with the club’s board in a bid to persuade tem to hang onto their star man.
The Argentine is desperate to keep his talismanic forward, who has spearheaded their run to the Champions League final and potential La Liga triumph.
Simeone is thought to want one more season from Costa to help him cement his team’s spot at the very peak of European football, and he will not stand in his player’s way if he still wants to move next summer.
But Chelsea are still confident that they will be able to lure the front-man to Stamford Bridge as the answer to Mourinho’s calls for a “killer” goalscorer.