The transfer is now set to be closed following Spain's elimination from the U21 European Championships.
Asenjo said: "Atlético is one of the big three of Spain with Barcelona and Madrid. It's a big dream for me to play there.
"Atlético are the only team that has made their interest concrete. I will meet my family and my representatives and rest for the next fifteen days before my future is clarified. Now what I want to rest and make a decision.."