And Costa could now complete the details of his move and be in Madrid before Chelsea face Atletico next week.
Spanish radio station COPE claimed a deal has been agreed worth an initial £48.8m with £8.8m in add-ons.
Costa remains in Brazil, AWOL from Chelsea, but may never move back to England.
The Spain striker has been told he has no future at Chelsea, despite failing to agree a move before the transfer deadline closed and being registered in Chelsea’s Premier League squad.
But the saga looks to finally be coming to an end.
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Chelsea signed Alvaro Morata as his replacement this summer and he’s started well but appears to lack the bite Costa gave them up top.
Morata has scored three times in five appearances and is a more elegant player, but lacks the fighting instincts Costa had.
Costa scored 20 goals as Chelsea romped to the title last season but was dumped via text this summer.
He has since refused to come back to London and has been fined by Chelsea every week.
Chelsea started the season poorly with defeats in the Community Shield and the opening game of the season before a change in fortunes.
Prior to the goalless draw with Arsenal on Sunday they had won four in a row and dispatched Nottingham Forest 5-1 in the Carabao Cup last night.
And the club will breathe easier once the Costa saga is put to bed finally, allowing them to move on from one another and focus on the future.