The Manchester City winger, who has two years left on his contract, faces an estimated seven months on the sidelines after tearing his anterior cruciate ligament in the Community Shield against Liverpool.
Sane is undergoing surgery in Austria this weekend, the Germany international opting to be treated by Professor Christian Fink rather than City's usual specialist, the Barcelona-based Dr Ramon Cugat.
Despite the 23-year-old being likely to miss most of the 2019-20 campaign, reports have suggested Bayern could still continue their chase for Sane even though Ivan Perisic has arrived on loan from Inter and Philippe Coutinho is set to follow from Barcelona.
However, despite the need to replace veterans Arjen Robben and Franck Ribery, Rummenigge has ruled out a move for Sane ahead of the Bundesliga's transfer window closing next month.
"No, guys. Everyone knows what a cruciate ligament tear means and what the downtime is," Rummenigge told reporters on Saturday.
Bayern have confirmed a deal in principle is in place for Coutinho to join on loan from Barca for the season, with the deal for the Brazil international including a purchase option.
Rummenigge, meanwhile, would not say whether Bayern intend to keep defender Jerome Boateng at the club.
"I cannot say anything conclusive today about that," he added. "The transfer market is open until September 2. And then we'll see how the whole thing develops."
Boateng was an unused substitute as Bayern opened their title defence with a 2-2 home draw against Hertha Berlin on Friday, when Benjamin Pavard and Niklas Sule started in central defence.
Bayern have also confirmed Javi Martinez, who was not available to face Hertha due to injury, returned to training for Saturday's session.