The 24-year-old was reported to be the subject of a late £22million bid by Brendan Rodgers in the closing stages of this summer's transfer window.
Costa, who started his career with Portuguese side Braga, conceded a move to Anfield was on the cards but ultimately chose to reward the loyalty Atletico have displayed to him.
"I was close to leaving, Liverpool are a great team," he told Mundo Deportivo. "But after fighting so hard and overcoming difficulties to get my place at Atletico, how could I leave?"
Liverpool have since been linked with a January bid for the forward, but Costa remains adamant that he will be staying at the Estadio Vincente Calderon, where he is relishing the UEFA Champions League challenge ahead.
"I thought it was very important to keep growing at the Colchoneros, and play here for many years," he added.
"Porto are a team with a lot of traditions in this competition, Zenit are a great team too. It's a difficult group, but we knew we wouldn't have an easy group in the Champions League, considering what could have happened, the group is good.
Despite Costa's two Brazil caps to date, there have been recent suggestions the forward could represent Spain in future internationals, speculation he is not denying.
"In my life, I usually follow this idea; I want to be in a place where people want me to be," Costa admitted. "I play where people want me, I'd have to see, I don't close any doors."