The Spaniard looks set to stay at Bayer Leverkusen after rejecting the Reds and Bayern Munich.
Alonso wants to hone his craft and may be holding out for the Real Madrid job in a year's time.
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Per The Times, the Reds have moved on and are looking at alternative options.
The likes of Sporting CP boss Ruben Amorim have been linked, while Brighton's Roberto De Zerbi is also mentioned.
Benfica boss Roger Schmidt is an outside candidate, while Lens manager Franck Haise is an option.