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Feyenoord head coach Arne Slot appears to be the new frontrunner.
Liverpool doesn't want a complete U-turn in the way that it plays.
Over the years, he has said all the right things when it comes to the biggest boxes that Liverpool needs to ensure are ticked with its new manager.
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Ahead of the Europa League Conference final against AS Roma two years ago, Slot told UEFA.com: "We are an attack-minded team that likes to have a lot of the ball, and we play with a lot of intensity.
The latter, in fact, according to Voetbal Zone, could take over at Feyenoord if Slot ends up at Anfield.
A few months later, after signing a contract extension with Feyenoord, Slot told the club's official website: "I feel that our work is greatly appreciated within the club and among the fans, not only because of the results but above all with regard to the development of players and the attractive, infectious way of playing the fans clearly enjoy."
Getting the most out of resources will be a key factor for Liverpool in deciding who takes Klopp's job and there is a clear synergy between the team and the fans in the Netherlands that can only be a big positive as well.
Slot's style and preferred way of playing, just like Klopp's, is ideal for fostering that.
While he is similar in many ways to Klopp, though, Slot has also made no bones about his admiration for Pep Guardiola.
"City has — and then it is not the intention to compare us with them — a similar way of playing how we want to do with Feyenoord: 4-3-3, building backwards, wanting to put pressure quickly."
It is interesting that he calls his set-up a 4-3-3 there — the same formation that Liverpool uses — when he has also used a 4-2-3-1 system at times.
Bayern Munich and Barcelona are both looking for new head coaches as well, but Slot has made it clear before that he sees himself in the Premier League at some point.
During his coaching career to date, Slot has said exactly what Liverpool would want to hear from a potential new manager.