The Bundesliga champions have already brought in Benjamin Pavard and Lucas Hernandez, the latter in a club-record €80 million (£72m/$90m) deal.
But further recruitments are required, particularly with Franck Ribery and Arjen Robben having departed, and Bayern have so far been frustrated in their pursuit of the likes of Sane.
And chairman Rummenigge remains confident, though, content to maintain a watching brief for the time being.
Asked specifically about top target Sane, Rummenigge told a news conference: "The status is that he comes back from vacation this week and did not want to talk during the vacation time.
"Now you will have to wait in peace and quiet to see if something moves in this direction.
"It is well known that we are still looking for a wide player. We lost Franck Ribery and Arjen Robben, two players who have shaped the face of Bayern."
The 63-year-old went on to hint that there will be plenty more activity in the market at the Allianz Arena before the start of the new season, despite intense competition for new signings.
"Internationally, the competition is greater and the market has become more global," Rummenigge added. "Nevertheless, Bayern still have much to offer. We have a beautiful city, a nice stadium.
"But you have to say the buzz is different. We are speaking in the nine-digit range almost as if it is normal.
"There are almost two months left until the transfer market closes, and these two months will bring many more uncertainties.
"Everything is waiting for the dominoes to fall and then there will be this effect. Then let's wait and see what Bayern will do."