The Reds were the better team in the first half before Mohamed Salah suffered a serious shoulder injury to force the Egypt international off the pitch.
Liverpool FC were 1-0 down at the start of the second half after Loris Karius’ mistake allowed Karim Benzema to break the deadlock in Kiev.
Senegal international Sadio Mane levelled before Gareth Bale came off the bench to score an incredible overhead kick and restore Real Madrid’s slender lead.
Liverpool FC conceded a third when Karius fumbled Bale’s long-range shot into the net to seal a 13th European Cup crown for Real Madrid.
The Reds relied on Jordan Henderson, James Milner and Georginio Wijnaldum in the Champions League final due to Alex Oxlade-Chamberlain’s injury.
Emre Can started on the bench but the Germany international is set to leave in the summer on a free transfer unless the Reds can convince the Juventus target to sign a new deal.
Former Liverpool FC defender Carragher wants to see Jurgen Klopp make improvements in midfield.
“Going forward I certainly think the midfield needs looking at,” Carragher said on the Telegraph’s podcast.
“That will change, with Keita coming in and talk of Lyon’s captain [Nabil Fekir] also coming in who plays that No 10 role.
“I think we lacked something in midfield I’d say.
“We got to the Champions League final with that midfield three of Wijnaldum, Milner, Henderson, they’ve been absolutely superb – but they’re workaholics.
“The class of Real Madrid in the centre of midfield with Kroos and Modric told really.
“We didn’t have a player who could match them on the ball. I think it’s important that we bring in a couple of bodies in that position. I’m sure that’s what the manager will look to do.”
Liverpool FC signed Fabinho in a £39m deal from AS Monaco in the summer transfer window.
RB Leipzig midfielder Naby Keita is set to link up with the Reds this summer.
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