Ancelotti left the Allianz Arena club last week after their 3-0 UEFA Champions League defeat to Paris Saint-Germain.
The Italian won the title in his first campaign with Bayern but appeared to pay the price for a poor start to the season, with the club now five points behind Borussia Dortmund in the Bundesliga table.
Bayern chairman Uli Hoeness has also said his position had become untenable as "there were five players against Ancelotti," per Westfalenpost (via The Independent's Jack Austin).
Opta noted his record at Bayern was bettered only by Pep Guardiola:
2.25 - Only Pep Guardiola won more points per competitive match as @FCBayern coach (2.39) than Carlo Ancelotti (2.25). Gentlemen. https://t.co/GjraPIQ5Un2017-9-28 14:13:45
Per Sky Sports Germany (via MailOnline's Will Griffee), Metzelder has said Klopp could do the job: "From my point of view, Jurgen Klopp would be ready. He is a top man-manager and could make the complex power relationship work wonderfully."
Klopp joined Liverpool in October 2015 after stepping down as Borussia Dortmund boss at the end of the previous season, and he signed a new long-term contract in the summer of 2016 that will keep him at Anfield until 2022.
Under Klopp's leadership, the Reds secured a top-four finish last season, ensuring a return to the Champions League for the first time in three years.
The German has also instilled an exciting, attacking brand of football, through the pace and guile of Sadio Mane, Roberto Firmino, Mohamed Salah and Philippe Coutinho.
Although Liverpool remain vulnerable defensively, as 12 goals conceded from seven Premier League games demonstrates, Klopp will surely want to continue building the team and continue the work he has already done at Anfield.
ESPN FC's Alex Shaw does not believe Bayern will move for Klopp at this time:
Alex Shaw @AlexShawESPN
Nagelsmann to Bayern in the summer, you'd think. Doubt they'd go for Klopp. And maybe Carlo back to AC Milan? https://t.co/x3yu6av1Mk2017-9-28 12:29:53
There are plenty of other names in the frame, with former Barcelona manager Luis Enrique reportedly set for talks with the club on Wednesday, per Sport1 (via Marca).
Former Manchester United manager Louis van Gaal has also been linked with taking over the job until the end of the season. Hoffenheim's Julian Nagelsmann would then be appointed as head coach on a permanent basis, per Westdeutsche Allgemeine Zeitung (via Mark Brus of Metro).
Per Abendzeitung (via Mark Lovell at ESPN FC) Hoeness has said he has already told Guardiola who will be appointed, suggesting the decision has been made.