The Champions League finalists have ramped up their pursuit of the Benfica attacker as they prepare to replace Sadio Mane, who is wanted by Bayern Munich.
Liverpool got an up-close look at the prolific Nunez in the Champions League this season and were impressed with what they saw, with the view being that he can make the grade at Anfield.
Manchester United are also interested in Nunez, who has attracted interest from a host of clubs, but Liverpool are poised to take an important step forward in their pursuit.
Liverpool are now close to agreeing a deal, with personal terms also on the table.
Once personal terms are signed off, Liverpool will be free to focus on finalising a fee with Benfica.
Benfica have been tipped to want around the €100m mark to let Nunez move on and it remains to be seen if Liverpool are prepared to smash their club record to land him.
They dipped into the Portuguese market in the January transfer window when they landed Luis Diaz from FC Porto and he has quickly adapted to life on Merseyside.