The Reds acted fast earlier this summer to land the Uruguay star in a deal which could reach a club-record £85million.
Many were impressed with the swift and easy nature of the move, especially given the 23-year-old striker was among the most sought-after players in Europe.
Nunez emerged as one of the continent's hottest prospects after dazzling for Benfica in the Champions League and the Primeira Liga.
West Ham saw a £45m offer rejected in January, with Benfica keen to see how the Champions League knockout stages would affect Nunez's price tag.
It is understood Man United also had a meeting arranged with the Portuguese club, but president Rui Costa had to cancel due to illness.
Now, transfer journalist Fabrizio Romano has revealed the true extent of interest in Nunez - with Arsenal, Newcastle and Real Madrid also keen.
Speaking on his YouTube channel, Romano said: "There were many clubs interested.
"In January, Arsenal and Newcastle made proposals for Nunez, but it was impossible for Benfica to sell the player in January.
"On Deadline Day, West Ham made a proposal, same answer from Benfica: 'No way will we let the player go now'.
"Also in June, Newcastle tried to jump into the race, they called his agents, but Darwin wanted Champions League football and he wanted Liverpool."
He added: "Real Madrid were very interested after missing out on Kylian Mbappe.
"But it was not possible due to the passport situation. Nunez has no European passport and they could not register him (having reached their quota of non-European players).
"Manchester United wanted him, with [manager] Erik ten Hag looking for a striker.
"Ten Hag was pushing for the deal, but United's budget was very clear 70m (euros), including add-ons, no more than this.
"Liverpool are going to pay 75m guaranteed fee with add-ons to arrive at 100m euros as a complete package.
"Man United decided to leave the race because it was impossible to get into a bidding war with Liverpool.
"Jurgen Klopp was the key man. He makes the difference for Liverpool.
"He called Nunez directly and the player was really excited by the prospect of joining Liverpool. Then they were really fast."