The Premier League duo were the last pair out in Friday's quarter-final draw in Nyon, which also saw holders Real Madrid given a tie against Juventus, last season's beaten finalists who knocked out Tottenham this term.
Anfield will stage the first leg of the all-English quarter-final on April 4, with the second leg to be played on April 10 at the Etihad Stadium.
The two clubs have already played each other twice in the English top flight this season - with the Reds inflicting City's only Premier League defeat so far.
City strolled to a 5-0 home win over Liverpool in September, with Gabriel Jesus and Leroy Sane both scoring twice after Sadio Mane had been sent off.
Liverpool gained revenge at Anfield in January when they inflicted what so far has been the only Premier League defeat for leaders City, who were beaten 4-3 and needed two late goals to add respectability to the final scoreline.
Manchester City director of football Txiki Begiristain was in Switzerland for the draw, and afterwards admitted the tie was "a very difficult one".
Klopp took questions on the European fixture as he spoke ahead of Saturday's Premier League home game against Watford.
The German said: "I think it is an absolute dream draw - for all Manchester United fans. I really don't mind.
"I knew it would be difficult and that is difficult. The good news is for Manchester City it will be difficult as well.
"I saw Txiki Begiristain's face after the draw - and it didn't look like it was Christmas and Easter on the same day."
Pep Guardiola's City side secured their place in the last eight by beating Basel 5-2 on aggregate, winning the away leg 4-0 but surprisingly then losing 2-1 in Manchester.
Liverpool saw off Porto in the round of 16 with an impressive 5-0 victory at the Estadio do Dragao and a goalless draw at Anfield.
Begiristain hinted an all-English tie had taken the shine off City's quarter-final draw.
"We like to travel and to go to another country in the Champions League but it is what it is," Begiristain said on City's website.
"We know each other very well, they are a very offensive team, the truth is it is going to be a great couple of games and we are going to see good football.
"We are doing well in the league but is the same as always, you have to be right, in the Champions League there is no time for mistakes, everything has to work, you have to be almost perfect in both games.
"We know the power and the strength of Liverpool - in the league they beat us, we beat them at home."
Elsewhere in Friday's quarter-final draw, LaLiga leaders Barcelona were drawn to face Roma.
Sevilla's reward for victory over Manchester United was a clash with German giants Bayern Munich.