Klopp announced his departure from Liverpool on Friday, but the club had known about the situation since a meeting in November.
The 56-year-old said in a video interview on social media that he is “running out of energy” and explained that club executives had already had time to digest his decision.
Liverpool were taken by surprise in the meeting in November — with the announcement coming less than two years after he signed a long-term contract until 2026 — but they now have time to discuss candidates internally.
The replacement shortlist is not yet publicly known, but Bayer Leverkusen manager Alonso, who is regarded as one of the most exciting young coaches in world football, is an obvious choice for the role.
Leverkusen are on course to win their first-ever Bundesliga title, having gone undefeated in their first 18 matches to build a four-point lead over Thomas Tuchel’s Bayern Munich.
Bayern have dominated German football for over a decade, last failing to win the league title when Klopp’s Borussia Dortmund side pipped them to glory in 2012.
There is no release clause in Alonso's Leverkusen contract but they are unlikely to stand in his way if he asks to leave at the end of the season.
Alonso distanced himself from the Liverpool job at a press conference held before Leverkusen face Borussia Monchengladbach on Saturday, but also did not rule out the possibility of taking over.
"Honestly, I don't have a direct answer [about the Liverpool job]," he said. "I'm happy here right now, that's for sure, and only thinking about Leverkusen."
Alonso added: "I was surprised by Jurgen's news.
"I have huge respect for Jurgen and Liverpool. At the moment I am really happy here and enjoying my work. Each day and game is a challenge. We are in an intense and beautiful journey here at Leverkusen.
"Speculation is normal, but I have great respect and admiration for what Jurgen has done at Liverpool in the last nine years, and how he's done it. He has a connection with the fans, he's done a great job, but my focus is here at Bayer Leverkusen.
"I have great motivation to do a good job here, I'm very happy with the team, with the club, with everything, so that's my focus. My focus isn't on May, it's on tomorrow ahead of our game against Gladbach and helping my players to be ready.
"But what I will say is it was surprising what happened at Liverpool with Jurgen leaving."
The former Liverpool midfielder is also a club legend as a key member of the 2005 Champions League-winning team, which sensationally beat AC Milan on penalties in the final in Istanbul despite being 3-0 down at half-time.
The former Spain international midfielder made 210 appearances over five years at Anfield.
Liverpool have announced that assistant managers Pep Lijnders and Peter Krawietz, as well as elite development coach Vitor Matos, will also vacate their positions at the end of the season, with Lijnders "keen to pursue his own career in management".
Lijnders is listed high among the bookmakers' favourites for the role, along with Brighton manager Roberto De Zerbi and Germany boss Julian Nagelsmann.
Liverpool will similarly have to replace outgoing sporting director Jorg Schmadtke, who also announced his departure on Friday.