A Champions League winner during his time as a player at Liverpool, Alonso had been quickly installed as the favourite to succeed Jurgen Klopp after the German made his shock announcement in January that he will leave at the end of the season.
Sporting Lisbon coach Ruben Amorim is the new front-runner with the bookmakers ahead of Brighton manager Roberto De Zerbi and Germany boss Julian Nagelsmann.
Alonso's decision to remain with Leverkusen, who are unbeaten in 38 games in all competitions this season, ends Liverpool's hopes his arrival would soften the blow of Klopp's impending departure
The Reds' fortunes on and off the field have been transformed during Klopp's eight-and-a-half years in charge. Liverpool won their first league title for 30 years in 2020, a year after lifting the Champions League. Klopp is on course to leave in a blaze of glory.