The midfielder has been linked with a move away in January.
"I read news that I had asked to leave in January, I laugh," said Burgui. "I would never ask for that, I am very happy in Gijon.
"I would stay here for a long time. It has cost me a lot of sacrifice to be where I am. I want to reverse my situation.
"I know I need to be better and I'm determined to prove myself."