According to 'Marca', the Spanish side are desperate to complete a deal for their former talisman and are prepared to pay a club record fee.
The deal would be made up of an estimated £48m up front, with a further £9m in add-ons.
Costa only has himself to blame for not finalising a move away from Stamford Bridge in this summer's transfer window. The striker's actions of refusing to return from Brazil and report for training are the reasons Chelsea played hardball over a potential deal.
The 28-year-old is still yet to return to London, putting his future at risk and making Chelsea less likely to accept the fresh bid.
The Blues were demanding £50m+ for their wantaway striker, meaning this offer could finally put an end to the prolonged transfer saga.
If the bid is rejected, however, the Spanish publication claims Atletico are unlikely to come back with an improved offer as they refuse to pay any more for a player Chelsea see as invaluable.
If Costa is to become an Atletico player, he will not be able to play for the club until January with the club under a transfer ban until January.
Though Diego Simeone is hoping Costa can start training before then so he can regain his match fitness, having not played a competitive game in 98 days.