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According to the Daily Mail, Liverpool are prepared to spend as much as £85m on the Uruguay international, who has also been linked with the likes of Manchester United and Atletico Madrid.
Should Liverpool pay that figure, it would be a new club record transfer – beating the £75m fee they paid to Southampton for Virgil van Dijk in 2018.
Liverpool's new sporting director, Julian Ward, has already entered into negotiations with Benfica to sign Nunez and the 22-year-old is reportedly keen to move to Anfield.
Nunez, who can play on the left side of the attack as well up front, scored 34 goals in 41 games last season which included two against Liverpool in the quarter-finals of the Champions League.
Liverpool boss Jurgen Klopp was impressed by the striker, saying in April: "A good looking boy! And a decent player as well.
"Really good, really good. I knew before, of course, but he played pretty much in front of me with his tough battles with Ibrahima Konate.
"He was physically strong, quick, was calm around his finish. Good, really good. I always say in these situations if he is healthy, it's a big career ahead of him."
Nunez is said to be "excited" to play under Klopp, with Liverpool moving ahead of Atletico Madrid and Manchester United in the race to sign the in-demand striker.
United boss Erik ten Hag is keen to sign the Uruguayan, who scored the goal that knocked his former club Ajax out of the Champions League last season.
But the striker, who has the same agent as Liverpool stars Fabinho, Diogo Jota and Fabio Carvalho, is keen to play in the Champions League again next season, seemingly ruling out a move to Old Trafford.
Should Liverpool pull off the transfer, it would be a significant statement of intent after narrowly missing out on the Champions League and Premier League trophies last season.
The Merseyside club have moved quickly to identify a possible replacement for Sadio Mane, who looks destined to leave Anfield this summer for Bayern Munich.
The German giants have already had one offer of £21m plus £4m in add-ons rejected by Liverpool for Mane, who has just one year remaining on contract.
Nunez first moved to European football in 2019, joining Spanish side Almeria from Penarol in Uruguay.
He later joined Benfica in a club-record €24m (£20.4m) move in 2020, with Almeria also entitled to 20 per cent of a future sale of the striker.
In his time at the Estadio da Luz, Nunez has scored 47 goals in 84 games.