Bayern’s interest in Sane is no secret, and Kovac said recently that he was confident about signing the 23-year-old.
But club chief Rummenigge is concerned that making a statement about a player under contract at City could harm relations between the two clubs, particularly with City boss Pep Guardiola vocal about his desire to keep him.
"I did not like what he said,” Rummenigge told ZDF.
“We have a good relationship with Manchester City and our former coach Pep Guardiola.
“Sane still has a contract there. Neither optimistic nor pessimistic quotes do anything to help us in that matter."
The Germany international has been in good form in pre-season but with the likes of Raheem Sterling, Bernardo Silva and Riyad Mahrez competing for his spot on the wing, it isn’t clear how much he would be likely to start for City in the coming season.
Having been a key man in the 2017-18 campaign, his surprise omission from Germany’s World Cup squad last summer was followed by a spell out in the cold at the Etihad.
Bayern, meanwhile, want to build around him as one of the biggest stars of German football, with Kovac describing him as a ‘dream player’.
The saga has been rumbling on throughout the summer. City would now have to act swiftly to replace Sane with less than two weeks of the English transfer window remaining.
The deadline on August 8 comes after the Community Shield against Liverpool, and Guardiola isn’t likely to be too keen on stretching his squad any thinner as they look to challenge on four fronts yet again.
While Sane didn’t play as much as he would have liked last season, his return of 16 goals and 18 assists in 47 appearances underlines why City want to keep him.
One of those goals, the winner in the 2-1 home win over Liverpool, could easily be labelled City’s most vital goal of the year. If that game had stayed at 1-1 and other results remained the same, The Reds would have won the league.
That sort of quality is why Bayern would like to get him in as soon as possible. Their season kicks off with the DFL-Supercup against Borussia Dortmund on Saturday.