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Yet both sides would be left disappointed when Alonso gave a press conference in which he affirmed his decision to remain at Leverkusen for another season, wanting to build on the success of this campaign.
This now means that Liverpool will have to focus on other targets to replace Klopp, with less than two months of the current campaign remaining and the German's time at the club coming to an end.
Given Alonso's decision, it remains to be seen whether he will ever manage Liverpool, or perhaps it's just been postponed to further down the line.
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Rúben Amorim is the new favorite to succeed Klopp, but there could still be plenty of time before we find out the identity of Liverpool's next manager.
Here's a round-up of the latest from the Reds search.
Paul Merson in shock at Xabi Alonso decision
Former Arsenal and Aston Villa midfielder Paul Merson has professed his shock at Xabi Alonso's decision to remain at Bayer Leverkusen next season.
Most expected the 42-year-old to leave Leverkusen after leading them to a probable league title — the first in the club's history — and sign for either Liverpool or Bayern Munich.
Alonso declared his intention to remain in the Bundesliga with Leverkusen this week, ending all further speculation linking him with the Liverpool job, and Merson is struggling to get his head around it.
They're two of the biggest jobs in the world of football," he added.
Liverpool.com says: Merson is right in that sometimes managers need to strike when the iron is hot, yet it's also admirable that Alonso rejected the chance to coach either Liverpool or Bayern, a decision he may or may not come to regret down the line.
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