The Uruguay international completed an initial £64m move to Anfield last month in a deal that could eventually see him become the Reds' most expensive player in history.
Nunez and Liverpool had already butted heads in the Champions League, with the former's Benfica side being eliminated by the Merseyside giants in last year's quarter-finals.
However, Nunez hit a goal in either leg against Klopp's side and also had a couple more disallowed at Anfield as Liverpool progressed 6-4 on aggregate before losing to Real Madrid in the final.
Klopp was full of praise for Nunez before the teams' Champions League encounters, and the German has now admitted that he "fell in love" with the striker last term.
"It was massively impressive when he played in front of us [in the Champions League]," Klopp told liverpoolfc.com. "When you prepare Benfica, you know up to the size of shoes - we know absolutely everything about an opponent, so we know exactly about the quality of Darwin.
"It was not clear if he would play because he came back from internationals, in the first game that was. It was clear that he was really good. But when you see it then live, it was really impressive.
"The power and the mix-up with technique, the desire, smart moves, the problems he caused us. We fell all in love in these two games. A very aggressive South American, it's a good thing!
'It was really a joy to watch him and so when we realised we had a good chance to sign him, we were all-in and we're really happy he's here now."
Nunez left Benfica after amassing 47 goals and 16 assists in 84 games for the club in all competitions since being signed for a £21.6m fee from Penarol in the summer of 2020.
The 23-year-old registered a staggering 34 goals in 41 games last term and was named the Primeira Liga Player of the Year after hitting 26 goals in 28 league matches.
Nunez will wear the number 27 shirt for Liverpool after signing a five-year deal until the summer of 2027 and followed Calvin Ramsay and Fabio Carvalho to Merseyside for the 2022-23 season.
Liverpool's first game of the pre-season period takes place against Manchester United in Bangkok on Wednesday before fixtures with Crystal Palace, RB Leipzig, Red Bull Salzburg and Strasbourg later this month.