Colombian midfielder Perez, 20, has spent some time with Arsenal this summer and played in their 3-1 pre-season friendly victory over Manchester City at the weekend.
However, he has not been granted a work permit and his club have confirmed that there will be no move to Emirates Stadium.
Asked about Perez's work permit situation, Atletico Nacional director Victor Marulanda is quoted as telling winsports.co: "We already knew that and were waiting for the response, but that wasn't possible and the player will keep working with us.
"The possibility of joining Arsenal has been ruled out."
Marulanda says Perez remains a transfer target for a couple of other clubs and will be allowed to go if a deal can be sorted.
He added: "There are two clubs interested in him, it's true. If there is any other concrete possibility, he will leave."