I think it's clear that Kane is more desirable to many clubs compared to Mbappe, because Kane's fee is lower than Mbappe and maybe it's also considered with their age. But regarding Bayern Munich's performance and squad, basically Bayern Munich will always be a strong candidate in the Bundesliga but not for the Champions League. Therefore, if Bayern Munich manage to sign Kane then at least Bayern Munich will have a better chance also in the Champions League, and not only in the Bundesliga. Therefore, it is not surprising that Bayern Munich want to bring in Kane, because it is hoped that the presence of Kane will make Bayern Munich stronger in every attack.
Bayern Munich still not open new proposal after first bid offer with 80 million euro denied by Tottenham Hotspur for signing Harry Kane, I think Bayern need push up their bid offering until 100 million euro if want sign Kane in this summer transfer. No doubt with Bayern Munich winning Bundesliga tittle without Harry Kane but has good chance in Champion League, however Tottenham Hotspur have give bid offer above 100 million euro for teams interested sign Harry Kane.
Bayern Munich is the only one teams have official bid offer for signing Kane, I think easy for Bayern if want push up transfer fees until 100 million euro because not any teams have offer bid yet although rumor Madrid and PSG are interested for signing Kane.