Costa, who cannot be officially registered by Atleti until January, has previously played in the same side as Luis at both the La Liga club and Chelsea.
Ahead of Costa's impending move to the Wanda Metropolitano, Luis hailed the world-class striker and his side's ability to pull off the transfer.
"I am proud that the club is doing all this," he told reporters. "An impressive stadium, wanting to grow more and more, to be up top, to fight for all titles. I say I am proud because a great effort has been made to bring one of the three best strikers in the world. Wherever he's gone he has made history, he has been champion.
"He comes to triumph and fight for something big and make us all better around him. I'm very glad that my friend is back here and now we welcome him in the best possible way. Any club in the world would not want to give up but Diego's desire to come was bigger than anything.
"Any club-player divorce is difficult, neither club were in a hurry to close the operation, so it has been delayed, I think, but in the end what we all wanted has been fulfilled. Diego returns home, returns to be happy, Chelsea are good with [Alvaro] Morata... Having Diego and letting him go shouldn't be easy."
Costa waves goodbye to Chelsea with 59 goals to his name, helping the club to two Premier League titles and one EFL Cup in his three seasons at Stamford Bridge.