The Uruguay striker is expressing a personal desire to return to Liverpool, though Mundo Deportivo reports that is not a realistic option and Chelsea represent his most advanced European interest.
Nunez joined Al-Hilal from Liverpool for €53m last August but was removed from their squad registration in February to accommodate the arrival of Karim Benzema — a decision forced by the Saudi Pro League's foreign-player limit. He made 22 appearances and scored nine goals before being effectively sidelined.
The termination by mutual consent leaves him free to sign for any club without a transfer fee — a significant attraction given his profile and age of 26. He earned £400,000 per week at Al-Hilal, meaning clubs will need to offer a substantial package to attract him but without bearing a purchase cost.
Nunez has been candid about his affection for Liverpool and the city. He told reporters earlier this year: "I'd be lying if I said I didn't miss them.
"The truth is, they made me feel very welcome, and my family too. During some difficult times I went through personally, the people always gave me that affection, that support."
Liverpool are not interested in a return. Incoming manager Andoni Iraola is pursuing other attacking targets and the club regard the chapter as closed.
Chelsea, who are building under new head coach Xabi Alonso, are monitoring the situation, as are Newcastle United and Barcelona — who regard him as a potential alternative to their primary striker target Julián Álvarez.