Both Arsenal and City were heavily linked with the Croatian in the summer before he left Sassuolo for Atletico, where he signed a five-year deal.
However, he has made just one La Liga appearance for his new side and there is already speculation about his future, with the Premier League touted as a potential destination.
Meanwhile, Napoli, who also tracked him in the summer, have been linked with a fresh move for him when the January transfer window opens.
But Vrsaljko says he is going nowhere despite his lack of game time in the Spanish capital.
“I’m not thinking of leaving Atletico,” he told El Mundo Deportivo.
“I do not want to talk too much about my situation, of course I want to play more, but things are the way they are.
“It depends on me working hard and showing the coach what I’m worth.”