Chelsea confirmed on Thursday that they had agreed to sell Costa back to Atletico for a fee believed to be around £50million.
Costa is due to arrive in Madrid later today for a medical and an announcement is expected over the weekend.
The 28-year-old’s career at Chelsea was ended via text message in the summer when Conte told the striker he was surplus to requirements.
The deal marks the end of three controversial seasons in English football for Costa but he leaves having scored 59 goals in 120 appearances and winning the Premier League title twice.
And speaking before boarding a flight to Spain, Costa stressed he was not upset at the way Conte had treated him.
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He told Splash News: “There’s nothing of the sort there, I’m not upset with anybody.
“I have no bitterness towards anyone. Everything is cool.”
He then sent a message to Chelsea supporters saying: “Everybody knows how much I love the fans.”
Costa refused to return to London, opting to remain in his native Brazil until an agreement was reached with the club he spent four years at before joining Chelsea in 2014.
The striker was warned by Spain coach Julen Lopetegui that he risked missing next summer’s World Cup if he didn’t play regularly.
Atletico are currently under a transfer ban from FIFA which prevents them from registering players until January and which means Costa’s second debut won’t happen until after the new year.
Phil Neville claims Costa is a better striker than his replacement Alvaro Morata.
He said: “I would rather have Costa than Morata in my team at this moment in time, to be honest with you.
“I think he is better suited to the English game – Morata has great potential, but if you were an opposition manager, you would rather be facing Morata than Costa.”