However, the 24-year-old, who was the subject of a £20m bid from the Premier League outfit in the transfer window, has claimed that he would have found it difficult to leave Atletico.
"I had the chance to leave. I was about to do so. Liverpool are a great team. But, after so much fighting and overcoming many difficulties to earn my place at Atletico, how was I going to leave?" he told Mundo Deportivo.
"I thought that it was very important for me to continue growing in this club and to play here for many years. Here I have the trust and respect of the people. My feelings towards Atleti? I know the world of football.
"I can really love a club, but a club is not just the fans, make no mistake. There are people running a company and with them you can encounter problems. The best way to demonstrate the affection or love that I have for Atletico is by giving my all on the field. That's my point of view. If I go, perfect. If I do not go, I try to improve."
Costa has scored three times for Atletico this season.