The veteran Spaniard came into the Anfield side from West Ham United in 2019.
However, he has been the second or third choice keeper for most of time at the club, only playing 26 games.
Confirming his exit on X, formerly Twitter, he said: "Dear Reds, I'm making this video to thank you from the bottom of my heart for everything you have given me during this beautiful and successful journey together.
"In five years you have shown me so many things. That wearing the Liverpool shirt and defending the Liverpool badge at Anfield should be added to the seven wonders of the world. That eleven yards is nothing when it comes to winning a title. That London is really Anfield South. That the Scouse accent is not difficult, it is for chosen and special people. That tradition must be protected. That You Will Never Walk alone is not just an anthem, it is a philosophy of life. Even that only three letters, K.O.P, can lift you up in the hardest moments.
"Now, it's time for me to share all these experiences I have learned. I will be grateful to you for my whole life. You have been and always will be very special to me and to my family.
"As Bill Shankly said, 'Liverpool Football Club exists to win trophies.' I wish you all the best. Once red, always red."
Adrian is now set to sign with Real Betis this week.