The 29-year-old spent the second half of last season on loan with Unai Emery’s side, scoring eight goals and providing one assist in his 21 games.
Aston Villa were understood to be keen on making the move a permanent one but were ultimately unable to get it over the line.
Asencio has now signed for Turkish giants Fenerbahce, putting pen to paper on a three-year contract with an option to extend it by a further 12 months.
Upon joining the club, he said: “I'm very happy to be here. I can't wait to play in front of our fans in this stadium.
“They told me there's a great atmosphere here. I want to join the team and train with my teammates as soon as possible. First of all, I'd like to thank them for their interest. I've received many messages from our fans.
“I hope to reciprocate their love by achieving great success together."