The winger left Manchester City in July 2020, but he has struggled to settle with the Bundesliga champions.
He’s yet to complete 90 minutes under new Bayern boss Julian Nagelsmann, and it remains to be seen how heavily he will feature in his plans.
Sane has started both of Germany’s World Cup qualifiers in the current international break, though, scoring in a 2-0 win over Liechtenstein.
According to the Express, Thomas Tuchel is a big fan of his compatriot and the Blues could look to land the 25-year-old in a swap deal.
Should that happen, the name which immediately springs to mind in terms of potential makeweights is Callum Hudson-Odoi – who Bayern had a bid turned down for last year.
Hudson-Odoi was linked with a loan move this summer – Atalanta and Bayern's Der Klassiker Borussia Dortmund both rumoured as possible destinations but ultimately stayed at Stamford Bridge, with Tuchel saying he believed the England winger wanted to fight for his place in the side.
Should Sane join the European champions - which isn't possible until January, of course - he'd link up with several familiar faces in the form of international teammates Antonio Rudiger, Kai Havertz and Timo Werner.
During his time at City, Sane made 145 appearances, scoring 39 goals and collecting two Premier League winners' medals.
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