Jupp Heynckes came out of retirement to take charge of Bayern until the end of the season after Ancelotti's sacking in September.
Sport Bild reported that Klopp is one of the names on Bayern's shortlist to become their next permanent head coach, with officials at the Bavarian giants considering the German attainable following the mixed start made by Liverpool to the 2017/18 season.
Bayern president Uli Hoeness has also said that it is "possible" that interim coach Heynckes could be appointed on a permanent basis at the end of the season, although the former Real Madrid coach has insisted that he has no intention of continuing at the Allianz Arena beyond next summer.
"I don't know why Uli said what he said," Heynckes was quoted as saying by ESPN last week.
"We have an extremely clear agreement and that runs until 30 June, 2018, and that is the way it is staying. I don't want to comment on this every week. That's pretty clear."
Thomas Tuchel, who is currently out of a job after he left Borussia Dortmund in May, is reported to be the third candidate under consideration.
Meanwhile, Bayern have cooled talk of approaching Joachim Low, who has a contract to manage the German national team until 2020, and Hoffenheim head coach Julian Nagelsmann.
Former Germany defender Christoph Metzelder urged Klopp to consider taking the Bayern job last month, saying he would be a perfect fit for the work culture at the Allianz Arena.
"He is a top man-manager and could make the complex power relationship work wonderfully," Metzelder was quoted as saying by the Independent.
"Nagelsmann is an outstanding coach but Bayern Munich is, of course, a different size [to Hoffenheim]. From my point of view, Jurgen Klopp would be ready."