The 32-year-old was spotted in Manchester's Bridgewater Hospital on Saturday, where he was said to be undergoing a medical ahead of a move to Manchester City's U.S. franchise.
"It is an irresistible project for me and my family , it is very, very good. It is about football and I have to accept it," Villa said in a message to fans published on the Atletico website.
"I will be a fan of Atletico all my life for what they have given me this year."
Villa moved to Atletico from Barcelona ahead of last season and leading the attack alongside Diego Costa, helped them win a first La Liga title since 1996.
They also came within 90 seconds of winning the Champions League before Real Madrid rallied to claim the title 4-1 after extra time in last week's final in Lisbon.
"It has been a great project for me both personally as well as sporting and now I face a new challenge which is what I have always looked to do in my career and my life," he added.
"I would like to thank the people at Atletico and thosethat tried to convince me to stay, like Cholo (coach Diego Simeone) and the rest of the coaching staff."
Villa was named in Spain's 23-man squad for the World Cup in Brazil, where the defending champions will open their Group B campaign against the Netherlands in Salvador on June 13 before facing Chile and Australia.
(Editing by John O'Brien)
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