Costa is set to be unveiled at Atletico after Chelsea and the Spanish giants reached a £67million agreement for the striker.
That version of events is disputed by Costa, who argued Conte was forcing him out after the Italian sent a text message saying the striker was surplus to requirements, despite having two years remaining on his contract.
With the personality clash becoming ever more acrimonious, Costa spent the last months of his time at Stamford Bridge in Brazil after refusing to return to train with the reserves as Conte wished.
"I'm not going to be ungrateful to Chelsea, where I was also very happy, playing in a team that was great", said Costa after arriving in Madrid.
Now Costa's future has been resolved, Chelsea can look to the future.
The player was in attendance at Atletico's La Liga game against Sevilla on Saturday afternoon.
Costa scored 52 Premier League goals for the Blues, helping them secure two titles in three years.
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"Don't forget we won together last season".
Costa, 28, joined the Blues from Atletico in July 2014, scoring 59 goals in 120 appearances and winning the Premier League twice.
However Costa will not be freed to make his second debut until January, as he waits for the next transfer window to officially open and for Atletico to finish serving a ban on registering new players.
Asked if he would shake his hand with his former player, manager Conte said: "Yeah, I have no problem". "You have to replace him and I think the club were very patient in the market and found the right player that they were looking for".
Atletico Madrid moved into the second place in LaLiga thanks to a 2-0 win over Sevilla at Wanda Metropolitano.
"Now we have to play three games".