Holders Liverpool, who won qualifying Group E ahead of Napoli, will have home advantage in the second leg against Diego Simeone’s Atletico, LaLiga runners-up last season.
The tie represents a return to the Estadio Metropolitano, the scene of the Reds’s June final win over Tottenham.
“Madrid is the place where we only have fantastic memories, all of us, so that’s great, but this time we play Atletico there and it’s their home ground,” said Klopp.
“I think we had the home dressing room in that game. Maybe we can ask them: can we have that dressing room again! It was really nice. It was full with Liverpool signs, but I think they’ve got rid of them.
“It is fantastic. What can I say? I had one of the best nights of my life in this stadium, but this time - first and foremost - we have to go there to work and try our best.”
Manchester City have a tough task in their quest under Pep Guardiola after being drawn to take on 13-time winners Real Madrid, but they too will have home advantage in the return leg.
Tottenham will play current Bundesliga leaders Leipzig and Chelsea face another German opponent in Bayern Munich, who won all six of their qualifying games in Group B.
Both Tottenham and Chelsea will travel to Germany for their respective second legs having finished runners-up in their qualifying groups.
City’s director of football Txiki Begiristain put on a brave face after learning his club’s last-16 opponents.
“It’s always a pleasure to play against Real Madrid, and also to play in a big stadium like the Santiago Bernabeu,” he said.
“We’re happy to go there, of course, and we know what we are going to face, we know them very well. Also, of course, they know us and what we’re doing and our manager and our team. Everybody knows each other, but it’s a real pleasure to go there.”
Chelsea’s meeting with Bayern will be a rerun of the 2012 Champions League final, which the Blues won on penalties in Munich.
Chelsea boss Frank Lampard was the club’s captain on that occasion, which he described as “the defining night in my career”.
But, on the upcoming games, he said on Chelsea Twitter’s account: “It will be very different and very taxing.”
Klopp, meanwhile, is hoping his rising young Liverpool stars are bursting with excitement and run for their lives as the Reds bid to cause a Carabao Cup shock.
With the bulk of Klopp’s first-team squad on World Club Cup duty in Qatar this week, Under-23 coach Neil Critchley will lead out a young, inexperienced side for the quarter-final clash at Aston Villa tonight.
Although Liverpool, unusually for the club this season, go into the game as underdogs, Klopp is far from ruling out the prospect of an upset.
“I know how a lot of people in the public see it,” Klopp told Liverpool’s official website.
“But if all the football games would be decided before we start, only because of the perception of the people, then most of the games wouldn’t be played. I believe, I’m a big believer in everybody has a chance.”