The 28-year-old was linked with a move to his former side all summer long after being informed by boss Antonio Conte in May that he is no longer wanted at Stamford Bridge.
A move failed to materialise before the window closed, however, leaving the Spain international - who has not played a single minute in 98 days - in limbo for the first half of the 2017-18 campaign.
Costa has repeatedly made clear that he will only leave West London if Atletico come calling, but club president Enrique Cerezo insisted last week that - contrary to press reports in Britain - a deal was not close to being struck between the two sides.
According to Marca, however, Atleti have stepped up their pursuit by informing Chelsea that their latest £57m offer is the last that will be tabled - £48m up front and £9m in various performance-related add-ons.
Atletico, who are under a FIFA-imposed transfer ban until January, had Costa on their books for a combined six years before he joined Chelsea in 2014.