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After going 2-0 up in the first leg at Anfield, the Reds were put to the sword by Madrid as Vinicius Junior and Karim Benzema each scored twice and Eder Militao also found the net.
It leaves Liverpool needing to pull off another Champions League miracle to reach the quarter-finals of the competition and keep their hopes of winning a trophy this season alive.
Liverpool has relatively recent history of overturning a three-goal first-leg deficit. It is one of just four teams to have progressed from a Champions League tie having lost the first leg by three or more goals, doing so in 2019 when it beat Barcelona 4-0 at Anfield in the semi-finals after losing 3-0 at Camp Nou.
Repeating the feat against Madrid at the Santiago Bernabeu is an even taller order.
Madrid has progressed from 26 of their previous 27 European Cup or Champions League knockout ties when winning the first leg away from home, with the exception being a defeat to Ajax at this stage in 2018-19 (2-1 away, 1-4 at home).
Klopp was keen not to raise hopes of Liverpool, who will be without Jordan Henderson due to illness, doing the seemingly impossible
"If there is only one per cent chance I would like to give it a try, we are here to play an extremely strong opponent and try to win the game," said Klopp.
"As difficult as it is, that's probably possible. Not likely but possible. We respect the competition too much, we respect the opposition too much to not look forward to this game."
Liverpool has endured an erratic season, losing eight games in the Premier League, six more defeats than they suffered in the entirety of last season.
In addition to shipping five goals to Madrid, Liverpool were also beaten 4-1 by Napoli in the group stage.
But, with Liverpool having also hammered Manchester United 7-0 nine days ago, it was put to Klopp that this strange season could be the perfect one in which to pull off such a turnaround against Madrid.
"If we can surprise ourselves in a negative way, we should be able to surprise ourselves in a positive way too," he responded.
"We are not here and telling Madrid 'be careful we are coming'. We're here to try to win the game. We have to play extremely good. If we play a normal game or a good game, it won’t be enough.
"I think in this room, 100 per cent of the people in this room think we have no chance. We are the complete outsiders. It's the situation, it's football and we try to give it a go."