The Portuguese club said that they had reached a deal with the English club.
The fee has been agreed and could rise to 100m euros with add-ons.
The Benfica player scored 34 times in 41 appearances.
£75m was paid to sign centre-back Van Dijk from Saints.
Premier League rivals Manchester United were reportedly also interested in signing Nunez, who cost Benfica 24m euros (£20.5m) when he signed from Spanish side Almeria in 2020.
He scored 26 goals in 28 league games in the 21st century, with 25 of those goals coming in the 24 league matches in which he started.
He scored six times in the 10 games he played in the European competition.
The Reds won the FA Cup and Carabao Cup last season, as well as finishing as runners-up in the league.
The first signing of the summer transfer window was a 19-year-old from Portugal, who is set to join from Cottagers on 1 July.
If an agreement over a new contract can't be reached, Sadio Mane will be a free agent next summer.
Brazil forward Roberto Firmino has entered the final year of his current contract and is attracting interest from a number of clubs.
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