Liverpool finished as runner-up to Napoli in its European group, allowing it to be paired with holder Madrid in the competition's last-16 draw on Tuesday (AEDT).
Klopp's side have lost two Champions League finals to Madrid in the past five seasons, but the German is looking forward to two "special" meetings with Carlo Ancelotti's men.
"I know we have played Real Madrid in a couple of recent finals, but it is not so often that our two clubs have met in a two-legged tie, so now we can look forward to this happening," Klopp said.
"Real's European record is the best around. We know this. But we also know that ours is not too bad. On top of this, we know that we have Anfield and everyone knows what this means.
Jürgen Klopp described our #UCL last-16 pairing with Real Madrid as a tie 'we should all be excited about' 👇— Liverpool FC (@LFC) November 7, 2022
"It is a really good draw, a really exciting draw. The games themselves are still a few months away, but it does not take a great deal of imagination to think what the atmosphere will be like at both games.
"There is a lot of football to be played before this, of course, so we will focus on each game as it comes. But yes, this draw has given us something special to look forward to."
Madrid has won all three of its UEFA Champions League meetings with Liverpool when facing it as European champion, and its former striker Emilio Butragueno believes Ancelotti's side will handle the pressure of being considered the favourite.
"I think that for the world of football, it will be an exciting tie because of the history of the two clubs," Butragueno said after the draw.
"They are a very strong rival, who have had a lot of stability in recent years with the same coach and with the same group of players, and that makes it very dangerous.
"At the same time, we are the champions, and we will naturally do everything possible to qualify and return to the draw for the quarter-finals.
"We fully trust our players, they have shown us what they are capable of in moments of maximum pressure."