They have seen him slip down the pecking order at the Etihad Stadium.
Former Bayern boss Pep Guardiola has been favouring other options on the flanks and the Blues have been unable to tie a proven performer to an extended contract as a result.
That situation could play into the hands of those at the Allianz Arena.
Bayern are in the market for more creativity in wide areas after seeing veteran performers Arjen Robben and Franck Ribery reach the end of their respective contracts.
Sane is the man they are prioritising, although Rummenigge admits there are no guarantees that a 23-year-old who has figured on the club’s radar for some time will be returned to his homeland.
Bayern’s CEO told Bild: "We'll try it. But I can't promise whether it succeeds.
"We already had the idea of signing him when he was at Schalke. But back then we had no guaranteed starting XI spot for him as Ribery and Robben were still at their top three years ago."
Pressed further on who Bayern will send to Manchester for discussions with Sane, Rummenigge said: "Hasan [Salihamdizic] will do that."
As they prepare to send their sporting director to England, Bayern will continue to monitor potential targets.
He has admitted to “dealing with this player” when quizzed on Sane, but added in Sport Bild: “I don't comment on names.
“We will try to do something for the new season, so we must also clarify how much money we have available for it.”
Bayern have already started reinforcing their ranks for 2019-20, with deals in place for French defensive duo Lucas Hernandez and Benjamin Pavard.
Jann-Fiete Arp has also been snapped up from Hamburg.