The Brazil-born frontman has been consistently linked with Stamford Bridge throughout this transfer window.
And despite assurances from Chelsea boss Jose Mourinho that Chelsea are not in the market for a striker, the report claims the Blues are ready to test Atletico's resolve with a dramatic late £30m offer.
The Metro notes that Arsenal and Manchester United have both scouted Costa in the last month, but Chelsea are prepared to make their move.
Mourinho clinched a deal for St Etienne defender Kurt Zouma earlier today, Chelsea's third deal of the window.
They're already signed Nemanja Matic and Mohamed Salah, but many prior to the January window expected Mourinho to move for a striker.
Fernando Torres is currently sidelined and Samuel Eto'o and Demba Ba have struggled for goals this season. That fact was underlined against West Ham on Wednesday night, when the Blues failed to break down the Hammers despite registering 38 shots.
And the report claims Chelsea may even include Thibaut Courtois or Ba as part of the deal to bring the Arsenal and United striker target over to Stamford Bridge.
United do not look likely to do any late business, according to David Moyes, who is happy with the club's £37m capture of Juan Mata from Chelsea.
Timing is running out for the Blues though, with Friday's 11pm deadline looming.