The 26-year-old came through the ranks at Sevilla before joining Liverpool in a £12m deal in 2014.
The left-back has spent the last five seasons at Liverpool but his contract expires imminently.
Moreno’s involvement under Reds boss Jurgen Klopp diminished drastically after Andy Robertson signed and took his position.
And he has now reportedly decided to quit the Premier League altogether and head back to Spain.
A source close to Turkish side Fenerbahce revealed that the club held talks with Moreno’s representatives.
But they were informed that he has his mind set on playing La Liga football next season.
The Mirror claimed the defender was on Arsenal’s radar, but he does not want to stay in England.
Arsenal’s current left-back options include Sead Kolasinac, who has been linked with a move to Barcelona and Nacho Monreal, who looks set to leave the club.
And now, www.Turkish-Football.com claim that Moreno is in talks with Spanish sides.
The reports adds that the star is close to an agreement with a team who will play Champions League football next term.
Assuming we can rule out Barcelona and Real Madrid, that leaves Atletico Madrid and Valencia.
Moreno will leave Liverpool as a free agent at the end of the month as the Reds did not offer him a contract extension.
Liverpool manager Klopp paid tribute to the Spain international’s efforts over his 141 Reds appearances.
He said: “The most important words to say to these two remarkable players are ‘thank you’.
“Alberto is a person whose personality is reflected in how he plays – full of life, full of energy, always positive.
“An unbelievable character who is always willing to give all of himself for the team.”