The 29-year-old’s current deal in Germany is set to expire at the end of the season with talks on going over a new contract.
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According to Sky Sports in Germany, Saudi side Al-Ittihad offered Sane a huge €100 million (£85m/$113m) contract for four years.
Sane has turned that offer down, however, in favour of staying Germany and is close to agreeing a new deal with the Bavarians.
The two parties are expected to reach an agreement that would keep him at the club until 2028 in the next couple of weeks.
Sane is likely to take a pay-cut from his current gross salary earning €10m per season, which could rise to €14m and €16m with bonuses.