Sane has struggled to get regular starts at City and could benefit from a return to the Bundesliga as Bayern look to replace Arjen Robben and Franck Ribery.
The German champions have reportedly made Sane a top target this summer, although Goal revealed last month that City still want to tie the 23-year-old down to a new contract.
But Rummenigge insists a deal could still happen if Sane shows he is keen.
"We have not made an offer for Leroy Sane. It is far from that," he told Sport Bild.
"First, the player has to decide if he can imagine coming to Bayern. If that's the case, then we'll go into concrete negotiations with Manchester City.
"The player is the most important factor. Apparently, Leroy has not yet decided. Hasan Salihamidzic [Bayern's sporting director] is leading the talks there.
"I think the player is exciting. Both in footballing quality and in terms of the type of player he is."
Bayern look set for an active summer of transfer activity with midfielder Renato Sanches admitting he could move on.
The Portuguese has endured a torrid spell since leaving Benfica in 2016 and wants to play more regular first-team football.
Speaking to Kicker, the 21-year-old revealed: "I said goodbye to Niko Kovac on Wednesday. Whether it was only for the summer break or forever, I do not know at the moment.
"I want to play more.
"I do not know if it makes sense to try again. I've played too little, I'm disappointed about that.
"Sometimes I asked myself, 'why is that?' All the young players like Joshua Kimmich, Serge Gnabry, Kingsley Coman play. Just not me. And I showed in training that I'm a good player.
"I think I have had the least chances of the young players.
"I do not want a new loan move. Should I leave, then I want it to be completely."