The top four sides in the EPL standings attain qualification for the UCL and Liverpool faces an uphill task to finish in the top four.
Mohamed Salah had opened the scoring after just three minutes, but Liverpool wasted a host of chances to extend their lead and paid in stoppage time.
Liverpool, who played out a 1-1 draw in the EPL against Leeds United were subjected to boos by the fans ahead of this particular fixture.
Liverpool stayed in sixth place, outside the Champions League places, after Spain defender Diego Llorente scored with a header from an inswinging corner to earn Leeds a point.
Madrid were far from their best, but did not need to be to set up a semi-final clash with Chelsea as Los Blancos extended their unbeaten run in all competitions to 14 games.
The talk about the match between Liverpool and Real Madrid has surrounded around their meeting in 2018 final in Kyiv, which Real Madrid managed to win 3-1.
Klopp is also now under the spotlight as his decision to make seven changes from Thursday's 1-0 defeat to Chelsea did little to aid his side's confidence.
Liverpool is currently at the seventh spot in Premier League standings and the side is four points behind fourth-placed Chelsea.
The loss of Champions League football would mean Liverpool take a significant financial hit on top of the ongoing effects of the coronavirus pandemic.
Former England and Manchester City goalkeeper David James believes players will have to agree with what the club requests them to do.
Liverpool has been decimated by injuries and they face a battle just to qualify for next season's Champions League via a top-four finish.
Curtis Jones dedicated his goal to absent Liverpool goalkeeper Alisson, whose father died in a drowning accident in Brazil earlier this week.
Whatever they tried, we had an answer: Jurgen Klopp 'very pleased' with Liverpool's win over RB Leipzig
Liverpool delivered an impressive performance to defeat RB Leipzig 2-0 in the first leg of their Champions League last-16 tie here on Wednesday.
Liverpool will square off against Premier League leaders City at Anfield on Sunday and the Reds have to win in order to remain in with a chance to lift the 2020-21 title.
Manchester City vs Liverpool: Jurgen Klopp, Pep Guardiola clash over rest time ahead of Anfield game
City had a game at Everton postponed on December 28 as part of a coronavirus outbreak that forced six players and staff members into isolation.