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Liverpool and Bayern Munich had both been keen on appointing their former midfielder but it appears it will be a losing battle.
According to the Liverpool ECHO, the Reds do not expect Alonso to leave Leverkusen this season and will now switch attention to alternative candidates, with Roberto De Zerbi and Ruben Amorim likely to be considered.
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De Zerbi will be at Anfield as early as Sunday when his Brighton side takes on Liverpool in the Premier League.
Liverpool has used data and statistics to put its next manager shortlist together and De Zerbi scores particularly highly, as does Amorim.
The Telegraph reports that the pair are the 'outstanding remaining candidates' for the job.
De Zerbi has also emerged as a possible target for Bayern too and Bild claimed earlier this week that the two parties had even held talks about taking over.
Journalist Ben Jacobs reports that the trio are all being considered and Liverpool hope to make an appointment before the end of the season.
Alonso meanwhile is set to speak to the media at his pre-match press conference on Friday.
According to journalist Guillem Balague, he will provide an update on his future and could announce his plans to stay with Leverkusen.
Liverpool.com says: There is no doubt that Liverpool has been considering its options and there will be alternatives to Alonso.
De Zerbi and Amorim are likely to be among them with the Brighton boss having an 'audition' for the role this weekend.