The Germany international has been strongly linked with a move to Bavaria this summer and Bild+ believes an official offer has been submitted.
Details are yet to emerge about the fee that was tabled by the Bundesliga champions but previous reports have suggested City’s valuation is around €90 million.
The news follows Bayern boss Niko Kovac’s comment that he felt “very confident” a deal would be struck for the winger.
Bayern CEO Karl-Heinz Rummenigge demanded an apology from the Croatian for those remarks, while sporting director Hasan Salihamidzic added: “We would be well advised not to say anything more about the matter, but rather to keep quiet both inside and outside.”
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