The madcap forward has been in his native Brazil for the last six weeks after being told he was surplus to requirements at the end of last summer.
Alvaro Morata has been signed as a replacement as Chelsea haggled with Atletico Madrid over Costa.
The Spanish side had only been prepared to spend £26m to bring him back to Madrid.
But following breakthrough talks they are ready to get much closer to the champions’ £50m valuation.
The Evening Standard say discussions in recent days have been extremely productive and a £41m bid is on the way which should end the saga.
Costa has made it clear he wants to return to the club he left for £32m in 2014.
And they are desperate to get him as well.
Atletico ace Koke has admitted he hopes Costa will soon be his team-mate once again.
He said: "We all want Diego Costa to come, but the situation is complicated.
"If Diego comes, he will help us and he already knows a lot of us at the club so for us it would be very exciting."
Atletico are under a transfer ban and will not be able to register players until January 1 2018 meaning Costa will not be able to represent his new club until next year.
