Real defeated Liverpool 1-0 in May to win Europe's elite club competition for a record-extending 14th time, having also beaten the Merseyside club in the 2017-18 final.
Liverpool manager Klopp, whose side qualified second from Group A behind Napoli and have struggled for form in the Premier League, said the match against Real was a "proper Champions League" tie.
"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 told Liverpool's website.
Liverpool, who host the opening leg at Anfield in February, won their sixth European crown in 2018-19 and only Real and AC Milan (seven) have won more. Bayern Munich have also won six.
"Real's European record is the best around. We know this," said Klopp. "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.
"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."
