The Reds drew 1-1 at Manchester United on Thursday night, 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 other ties see Braga of Portugal play Shakhtar Donetsk and Villarreal meet Sparta Prague while there will be an all-Spanish clash between Athletic Bilbao and holders Sevilla, who are chasing a third straight tournament success.
The quarter-final ties will be played on April 7 and 14.
THE SCOTSMAN ON SOCIAL MEDIA