The Reds will take on their arch rivals after the draw was made in Nyon on Friday.
The sides have never met previously in European competition, but will now meet for a place in the quarter finals in what is undoubtedly the tie of the round.
The first leg will be played at Anfield on Thursday 10 March, with the return at Old Trafford on 17 March.
Klopp’s side saw off German outfit FC Augsburg 1-0 on aggregate in the last 32, while United beat Danish side FC Midtjylland to seal their progress.
Louis Van Gaal’s side have already beaten the Reds twice in the league this season, winning 3-1 at Old Trafford in September, and then 1-0 at Anfield in January.
Liverpool will have no game in between the two Europa League fixtures. United will be in FA Cup action against West Ham.
Elsewhere in the draw, Tottenham will take on Borussia Dortmund, while defending champions Sevilla face FC Basel. There is an all-Spanish tie as Valencia play Athletic Bilbao.
Shakhtar Donetsk v Anderlecht
FC Basel v Sevilla
Villarreal v Bayer Leverkusen
Athletic Bilbao v Valencia
LIVERPOOL V MANCHESTER UNITED
Braga v Fenerbahce
Borussia Dortmund v Tottenham
Lazio v Sparta Prague