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The possibility of a transfer elsewhere only continues to grow as the days go by, with the club's focus expected to turn to other positions if they are able to wrap up Florian Wirtz's signing.
As the ECHO understands, it seems most likely that the Reds and their No.9 of three years will decide to part ways.
Here are the latest reports on Nunez and his potential exit.
READ MORE:Alan Shearer fires warning to Liverpool's rivals as talks continue over club-record transfer dealREAD MORE:Andy Robertson gives Florian Wirtz and Milos Kerkez Liverpool transfer verdict and contract update
Football Italia claim Serie A champions Napoli have made contact regarding the transfer of Nunez this summer.
Coach Antonio Conte has been promised signings in return for staying with the club, and a new striker appears high on the agenda.
Liverpool's No.9 is regarded as a 'strong' option on a list of candidates drawn up, which includes Lorenzo Lucca and Rasmus Hojlund.
Whilst recognising rival suitors from Saudi Arabia would have the upper hand financially, the Italian club hope that Champions League football and the chance to win another major European league are enough to sway his mind in their favour.
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Atletico Madrid have also been reported as another club who would hope to make the same case.
The Athletic claims that Al-Hilal has been informed of the conditions for Nunez's transfer.
TheSaudi Pro League side made contact on Monday.
The ECHO reported in January that representatives from the Gulf state had made contact with the Reds during the latest January transfer window.
The transfer window has come to a temporary close now, so competitors, such as Al-Hilal, are unable to register any more new players for the group stage.
The window reopens on Monday, June 16, however, and they will be able to sign players to register between June 17 and July 3 in time for the knockout rounds, provided they progress that far.
Nunez arrived from Benfica in 2022, signing a contract at Anfield through to 2028.
This is completely new to me and I’ve said many times that I’m working at a club here where it’s really nice to work.
"So mainly sporting directors are judged by the players they bring in, for me it’s very important to have a great working environment and I have that with Richard [Hughes].
"The last, last, last, last, last thing he would do, as I expect working with him for 10 months, is tell me: ‘If you play him, it would cost us this much’.
"That, he would never, never, never do.