Kane, 33, and Olise, 24, are among the most sought-after attacking players in European football, and Bayern are keen to avoid losing either as they continue building their squad.
Kane’s situation is particularly pressing because his current contract is approaching its expiry, while Olise is already tied to the German champions until the summer of 2029.
Rummenigge, a former Bayern chief executive who remains on the club’s board, told German outlet Ran that he expects progress with both players.
“Both of them intend to stay at Bayern for a long time. I hope — and I am also completely convinced — that Harry will extend his contract soon. Michael will be next,” he said.
The comments suggest Bayern believe the players are receptive to staying, although no new contracts have been officially announced.
Rummenigge also stressed how important it is for the club to retain players of their quality, pointing to the difficulty of replacing attackers who can influence matches at the highest level.
“At Bayern Munich, we can consider ourselves very fortunate to have two players — Harry Kane and Michael Olise — among the leading candidates for the Ballon d’Or,” he said.
“They are world-class players, and you do not find them growing on trees. They have been very important to us for some time.”
Kane’s contribution has made him central to Bayern’s plans since his arrival from Tottenham Hotspur. His goals and performances have strengthened his position among Europe’s leading forwards, and Rummenigge believes the England captain has a particularly strong case for the individual award this year.
“The decisive criterion is individual performance, and Harry Kane is ahead at this level. His goals and trophies speak for themselves,” he said.
Olise has also become an important attacking presence since joining Bayern, and the club’s desire to extend his stay reflects the value they place on his performances and future potential.
Securing new agreements would provide Bayern with greater continuity in attack and reduce uncertainty surrounding two players who attract attention from major clubs across Europe.
For now, the club’s confidence is clear, but negotiations still need to produce signed agreements. Bayern will hope the positive signals from Rummenigge translate into extensions for both players, beginning with Kane.