Benfica forward Nunez, who scored 34 goals in all competitions this season, has been touted as a possible replacement for Sadio Mane.
Mane looks increasingly likely to be heading for the exit at Anfield with Bayern Munich making an opening bid worth £25million, which was immediately rejected.
Liverpool may be holding out for a higher fee in order to fund an incoming player, who could be the £68million-rated Nunez.
talkSPORT's transfer expert Alex Crook said: "We spoke about Liverpool earlier, about Sadio Mane and would they replace him.
"Darwin Nunez, there is genuine interest from Manchester United, but a deal for him is not as close as has been suggested in the Portuguese media.
"I received a text message from a trustworthy source on Friday that Liverpool are genuinely contenders for him."
Former Liverpool star Danny Murphy has been impressed with what he has seen from the 22-year-old Uruguayan and believes he fits the Klopp mould perfectly.
Murphy told the White and Jordan show: "I watched him three times in the space of six weeks. One thing I really liked about him was that he is dynamic.
"He's a very Klopp-type player with high energy, both with and without the ball. He doesn't mind a press and saw him running back a few times.
"When he hasn't got the ball, he doesn't mind running in behind. He's more of a number nine, he's not going to play out wide."
Nunez has plenty of admirers in the game and even from current Liverpool players who have come up against him.
He played against the Reds twice in the Champions League this season, scoring two goals as Benfica were beaten in the quarter-finals.
When asked the toughest opponent he has played against, Liverpool centre-back Virgil van Dijk told Rio Ferdinand: "I've played against Sergio Aguero, who is obviously unbelievable. [Lionel] Messi, arguably the best player in the history of football, in my opinion.
"[Kylian] Mbappe is different quick. It's a different type of striker we face who is doing well in the name of [Erling] Haaland, also Darwin [Nunez].
"He's a bit similar, very direct and quick, tall and strong. It's quite difficult to play against those guys."