The 29-year-old is out of contract at the end of the season and reports are suggesting the winger is set to become a free agent.
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Both Liverpool and Arsenal are understood to be interest in Sane as they seek to bolster their attacking line-ups.
As per Sky Germany, Bayern have offered him a new deal that would keep him at the club until 2028, but he would have to agree to a significant wage drop.
Sane currently earns €20 million-per-season with the Bundesliga leaders, and he would see his salary drop to €10m and around €15m if he meets his performance-based bonuses.
It’s understood that Sane would prefer to stay at Bayern despite the proposed reduction in his wages and is not open to a move back to the Premier League.