Jupp Heynckes, who stepped in until the end of the season to replace Carlo Ancelotti in October, is also said to be in the frame to take the job full time.
Sport Witness relayed how Bild presented the story, with each of the four supposed candidates for the full-time position pictured:
SportBild's Christian Falk posted the following on Twitter, suggesting Bayern would swoop for Klopp if Liverpool axed him:
Klopp previously worked in the Bundesliga with Mainz 05 and then with Borussia Dortmund, with whom he memorably won back-to-back German titles.
He also led BVB to the final of the UEFA Champions League in 2013 before joining Liverpool in 2015. While he's yet to win silverware at Anfield, Klopp's side play some of the most exciting attacking football Europe.
With his pedigree in mind, it would be no shock if Bayern did attempt to prise Klopp away from Merseyside. And while Falk talks of Liverpool potentially disposing of the German, there doesn't appear to be any sign the club is dissatisfied with the job the 50-year-old is doing.
If he were to be sacked, The Anfield Wrap's Paul Senior suggested Klopp would be a peculiar choice for the Bavarian giants:
Low will be making preparations for next summer's FIFA World Cup, and holders Germany are among the favourites win the tournament.
The 57-year-old has done a fine job with the national team, overseeing an incredible rise. The end of the World Cup would see him round off 12 years in charge of Germany, and the chance to go into club football with Bayern could be an exciting new challenge for him.
Tuchel, meanwhile, is out of work after leaving Dortmund in the summer. The Scouted Football Twitter account believes he would be worthy of a job at a massive club:
If Heynckes does another good job at the Allianz Arena, there will be a temptation from those in high places to keep him in the hot seat for a while yet.
Even so, at 72, it would be a surprise if Heynckes envisaged himself taking charge for the long term and potentially even beyond this campaign. Bayern will be keen to have a plan in place and soon appoint a figure who can offer the kind of leadership a club of this size requires.