Tuchel left Dortmund last May and has been biding his time for the right opportunity.
Paris Saint-Germain and Chelsea have also been linked with the 44-year-old.
And he turned down the opportunity to talk to Southampton before they appointed Mark Hughes two weeks ago.
Sky Sports add if Tuchel does not get the Bayern job, he is open to the idea of a move to the Premier League.
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Arsene Wenger is under mounting pressure at Arsenal
But that he would only be prepared to join a club that he believes are among Europe’s elite.
Last week German football legend Lothar Matthaus suggested he felt Tuchel was on his way to the Emirates.
"I think Thomas Tuchel can be the favourite, [the] ex-coach from Borussia Dortmund, but I know he's in touch with Arsenal and nobody knows what happens behind the doors,” he said.
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