Sevilla, bidding to win the trophy for the third successive season, will tackle La Liga rivals Athletic Bilbao while Villarreal take on Sparta Prague and Braga face Shakhtar Donetsk.
After Liverpool knocked out bitter rivals Manchester United on Thursday, Klopp told reporters: "I hope we don't get Dortmund".
The charismatic German spent seven seasons at Dortmund, leading them to two Bundesliga titles, German Cup glory and the 2013 Champions League final where they lost to Bayern Munich.
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"He comes back!", said Dortmund playmaker Ilkay Guendogan on Twitter after hearing news of the draw.
Liverpool, without a major trophy since they lifted the League Cup in 2012, will travel to Germany for the first leg on April 7 before hosting the return match at Anfield seven days later.
It is 50 years since the two teams met in the final of the now-defunct European Cup Winners' Cup in 1966, Dortmund winning 2-1.
The Germans reached the last eight by defeating another English club, Tottenham Hotspur, in the last 16.
Sevilla have won their last two La Liga fixtures against Bilbao 2-0 but the teams have never met before in European competition.
Villarreal and Sparta have also never been drawn together before while Ukrainians Shakhtar, who won the UEFA Cup in 2009, beat Braga of Portugal twice in the group stage of the 2010-11 Champions League.
(Writing by Tony Jimenez,; Editing by Ed Osmond)
