The 22-year-old, who was left out of Uruguay’s squad for Saturday’s 5-0 friendly victory over Panama, was expected to begin a medical on Monday which could take two days to complete.
Personal terms are not anticipated to be a problem, with the player having expressed his desire to play for Reds manager Jurgen Klopp despite interest from Manchester United.
Nunez is set to sign a reported six-year contract with Premier League and Champions League runners-up Liverpool, who have been in talks with Benfica for several weeks.
He netted in both legs of Benfica’s Champions League quarter-final defeat against Liverpool in April, earning praise from Klopp.
Nunez is likely to arrive at Anfield as a direct replacement for Sadio Mane after Bayern Munich stepped up their pursuit of the Senegal forward.
Liverpool have rejected two offers from the Bundesliga champions and are holding out for a fee in excess of £40 million for Mane, who has one year left on his contract.