Atletico had refused to go above £35m for Costa, safe in the knowledge they are the only club he wants to sign for.
But with Chelsea refusing to budge on their valuation and Costa growing increasingly frustrated, the Spanish club have finally returned with a new offer.
Atletico have proposed an initial fee of £40m with the rest in add-ons, with the only major stumbling block being how those are paid.
The Madrid club’s president Enrique Cerezo said: “Am I pessimistic or optimistic regarding the arrival of Costa? I am always optimistic.”
Chelsea’s possible starting line-up after deadline day
CHELSEA FCBut the Spain star’s camp are likely to back down once Chelsea have agreed to his transfer.
There is also disagreement over who pays Costa’s fine for refusing to return to training with Chelsea, but this is not expected to scupper the deal.
Atletico have until midnight on Friday to sign Costa because the Spanish transfer window closes a day later than the UK’s.
But Costa cannot play for them until January when their transfer embargo is lifted, raising the prospect of him moving to a third club on loan in the meantime.
But Costa has made it clear he would rather spend the next few months training in Madrid than playing elsewhere.
Conte wants to bring in long-term target Fernando Llorente from Swansea to add to his forward options, and the club have submitted a £10m bid for the Spaniard.
But Chelsea are making contingency plans should they fail to sign Alex Oxlade-Chamberlain, who insists he wants to join Liverpool.
The Blues yesterday made an enquiry about versatile Bayern Munich wing-back Rafinha. Southampton’s Cedric Soares is also on their radar.
Chelsea also remain confident of landing Leicester’s Danny Drinkwater after the England midfielder handed in a transfer request.
And they are ready to make an improved offer for Everton’s Ross Barkley after having an initial £25m bid rejected.
