The Reds drew 1-1 at Manchester United, which was enough to see them progress at the expense of their Premier League rivals, while Dortmund defeated Tottenham 5-1 on aggregate to progress.
Klopp left Dortmund in July 2015, having taken the Bundesliga side to the 2013 Champions League final at Wembley.
The quarter-final ties will be played on April 7 and 14 (8 April and 15 April AEST).
"I am really happy with the draw," Klopp told Liverpool's official website. "Yesterday I said why should I wish [to get] the strongest team in the tournament, but obviously if you want to win the Europa League, and when you are in the quarter-final, then you have to beat the strongest teams. It is better to play against them in two games than in only one game."
"I loved this place, a few of the best things in my life happened there, so of course I am looking forward to it. But on the other side, I hate the hype around my person in these matches. I know about the business and all the circumstances, that's clear, but I don't like it to be honest.
"I will do what I have to do and then we will see. The game and the players, completely focused – or as much focus as possible – on all these guys. During the game, all the focus on these guys and then if somebody wants to talk to me after the game, I am available then.
"And then, they are all 100 per cent looking forward to coming to Anfield. It's football at its best and how it should be. Everybody is looking forward to it."
The other ties include an all-Spanish clash between Athletic Bilbao and holders Sevilla, who are chasing a third straight tournament success.
Meanwhile, Braga of Portugal play Ukraine's Shakhtar Donetsk while Villarreal will meet Sparta Prague.