Skysports and daily Bild say the clubs have agreed on a transfer fee of 32 million euros ($33.5 million) plus up to nine million in bonuses.
Mane is set to receive a three-year contract.
The 30-year-old could replace Bayern’s Poland star striker Robert Lewandowski who has repeatedly said he wants to leave and has been linked to Barcelona.
Mane would be Bayern’s third signing this summer after Netherlands midfielder Ryan Gravenberch and Morocco right-back Noussair Mazraoui were both signed from Ajax.
AS Monaco have confirmed Cesc Fabregas’ departure from the club on a free transfer.
The former Arsenal and Chelsea midfielder had been at the French side since the start of 2019 and made 68 appearances for the club – scoring four times.
Monaco also announced former Arsenal goalkeeper Vito Mannone’s exit at the end of his contract on Friday.
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