Bayern, who won their seventh straight German league title along with the German Cup this season, have repeatedly said they would like to lure the speedy 23-year-old winger to Munich as they look to overhaul their squad.
On Tuesday the club confirmed the widely expected departure of midfielder James Rodriguez, on loan from Real Madrid, who joins Frank Ribery, Arjen Robben, Rafinha and potentially Jerome Boateng on the way out.
But Rummenigge said reports of an 80 million euros offer for Sane were wide of the mark.
"We have not yet submitted an offer for Leroy Sane," Rummenigge told Sportbild magazine. "We are still a long way from this point.
"Initially the player has to decide if he could imagine joining Bayern. Once that is the case then we will go into specific negotiations with Manchester City."
After being voted the PFA's Young Player of the Year for 2017/18, Sane struggled to hold down a regular starting spot at City last season, competing with Raheem Sterling and Bernardo Silva among others.
City coach Pep Guardiola has said he had tried to extend the player's contract over the past year and a half but Sane's reluctance to put pen to paper has raised doubts about his future in Manchester.
"The player is the most decisive factor," Rummenigge said.
"Obviously Leroy has not yet decided. I find him an exciting player both in relation to his football qualities and his character."
Bayern have already signed defender Lucas Hernandez from Atletico Madrid for 80 million euros as they look to become more competitive in Europe, with their last Champions League win back in 2013.
They also opted against making the Colombia international Rodriguez' move from Real permanent after a two-year loan in Munich.
Rodriguez scored 15 goals and set up another 20 in 67 matches in all competitions for Bayern but struggled to earn playing time under current coach Niko Kovac.
The club said in a statement the 27-year-old had asked Bayern not to activate the buyout clause.
(Reporting by Karolos Grohmann; Editing by Peter Rutherford and Christian Radnedge)
