After going through the 2022-23 season without a direct replacement for Robert Lewandowski, Bayern managed to prise Harry Kane away from Tottenham Hotspur following protracted negotiations with Daniel Levy.
The England captain has already amassed 21 goals for the season during an exceptional start to life in Bavaria, where 18-year-old Mathys Tel has also scored six times so far this term and is being tipped for a bright future in the Bundesliga.
However, Thomas Muller and Eric Maxim Choupo-Moting - both of whom are 34 years of age - are due to leave the club when their contracts expire at the end of the season, while Serge Gnabry's future has also been called into question on numerous occasions.
As a result, Thomas Tuchel's side could try to bolster their attacking ranks once again when the summer window rolls around, and according to SPORT1, consideration is being given to a move for Mbappe.
Tuchel coached Mbappe during his spell as PSG head coach, and the German recently joked that he would pick up the striker himself if he showed a desire to play for the Bundesliga champions.
"Mbappe free next year? Yes, He will play for us. Kylian is Kylian, he is very intelligent," Tuchel told Canal+ when asked about the prospect of a reunion with the World Cup winner.
"We had a good relationship. If he wants to come to Bayern, I will go and get him myself. When he wants something, he gets it."
While Tuchel's comments were tongue-in-cheek, the report states that the Bavarians' board have not ruled out the prospect of an approach, especially with Mbappe due to become a free agent in 2024.
The France international was briefly ostracised from the first team this summer after informing PSG that he would not extend his contract beyond the end of the season, but he was quickly welcomed back into the ranks after productive talks with the club.
The striker is continuing to run down the final several months of his Parc des Princes deal, though, and despite rumours of an extension, nothing has been agreed with the Ligue 1 champions.
However, a separate report has claimed that Mbappe is after an extortionate yearly wage of €35m (£30.6m) from potential suitors, as well as a signing on bonus of €100m (£87.4m).
Real Madrid are understood to have definitively ruled out a move for Mbappe on account of his financial demands, while Liverpool - who have been tipped to rekindle their interest - would not sanction such a deal either.
Bayern would also need Mbappe to accept a slashed salary if he was to make the move to Germany, as the Bavarians have set a first-team wage cap of €25m (£21.9m) per year, which they do not intend to break.
Tuchel's men sit second in the Bundesliga table with 29 points from 11 games - two behind leaders Bayer Leverkusen - while Mbappe already has 15 goals from 15 games across Ligue 1 and the Champions League this season.