The Reds were hoping to appoint their former player in the summer, but that path is now unclear.
Per The Mirror and various other sources, Liverpool believe Alonso will be staying at Bayer Leverkusen.
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The young Spaniard has not only rejected the Reds, but also Bayern Munich.
Alonso may be more interested in waiting for the Real Madrid job, which is not vacant at present.
"We'll have to see if we can do it this year," said Bayern supremo Uli Hoeness recently.
"It will be difficult, if not probably impossible. (Alonso) is more inclined to stay at Bayer Leverkusen in view of their current successes, because he would not want to leave them behind.
"Let's say if he had two or three more years of success, it would probably be easier to bring him out of there."