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Should the Reds be unable to overturn their first-leg deficit against Atalanta on Thursday night, Klopp will have just six more matches left in charge.
There remains a chance that he could end his tenure on the ultimate high of winning the Premier League title, although Liverpool will likely have to win every remaining game, while also relying on Manchester City slipping up.
Speculation has been rife over who will replace him ever since he announced his plans to depart back in January.
Xabi Alonso was the early favorite but he will stay with Bayer Leverkusen, having just guided the club to the Bundesliga title.
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The new frontrunner is Sporting CP boss Rúben Amorim, who is also on the verge of winning the title in Portugal, but reports of an agreement being in place have since been dismissed.
And while Amorim appears an outstanding candidate, Liverpool is continuing to carry out extensive research on a number of options.
One of those is Germany boss Julian Nagelsmann, who will be out of contract after this summer's European Championships on home soil.
German FA chiefs have made no secret of the desire to keep him but there is a feeling he may be seeking a return to club management.
He is on Liverpool's radar, per The Athletic, but there is likely to be more competition from his former club Bayern Munich.
It is claimed that negotiations have been led by new Bayern sporting director Max Eberl.
Bayern is believed to be considering Brighton boss Roberto De Zerbi, another name linked with the Liverpool post, having also been interested in Alonso.
Unai Emery is said to be the final option on the shortlist.
The report comes after Germany FA director Rudi Völler revealed Nagelsmann had received approaches about returning to club football with a decision expected to be made about his future in the coming weeks.
Liverpool.com says: Nagelsmann is a talented coach and was harshly treated by Bayern Munich, a fact that Bundesliga outfit seems to have realized.
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