The move, thought to be worth around £18million, will take place when the transfer window reopens in July.
As reported earlier this week, Brazilian Costa is seeking a move to the Premier League and claimed Chelsea are his first choice.
Speaking to FourFourTwo, he said: "I'd be really interested in moving to Chelsea - not only because everything (midfielder) Willian tells me about the club, but because I think that I'd be able to achieve everything I want having someone like (manager) Jose Mourinho by my side."
"Since I left Brazil, I have worked with a marvellous coach [Mircea Lucescu] and got an idea of how much influence such a good professional can have in your football.
Douglas Costa Player Profile
"Now I think it's time to try a new philosophy and I'm sure that Mourinho's is the best around He's always fighting for trophies.
"Besides that, I'd have the chance to play in the Premier League, the best league in the world."
And that deal appears to have moved a step closer to being complete, report The Sun on Sunday.
Chelsea were interested in signing Costa, 24, in January but instead landed Fiorentina's £26.8m man Juan Cuadrado when Shakhtar rejected their advances.
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