This may well be a very outside and unpopular opinion, but
Andre Villas-Boas will be available come the end of next year. He's one of my personal favourite managers and I would very much welcome him (he did come to our Qatar camp 2 years ago didn't he?).
Now hear me out. I think we've established that Klopp is an exceptional manager whose record speaks for itself but just does not fit with our philosophy (or our vision towards a philosophy), and AVB is quite the polar opposite, being relatively inexperienced but does have a style about him. His managerial approach is personal and he likes to win with style and attack, attack and attack; something very reminiscent of our current coach. He's the sort of coach you're looking for if you're not only looking to win, but to dominate. He has a very clear vision when establishing his tactics, nitpicks in player positioning and sticks to it but can go very wrong on a bad day (again, who does this remind me of?). His playing style is not too pass heavy, not to the point where we want to have as many players as possible surrounding the ball so that the ball doesn't get lost (unlike Pep). But he clearly aims to keep possession and ideally making the opponent helpless whenever the team have the ball. As a bonus,
he attacks referees as well.
I don't think he's had the platform to truly establish himself as a world class manager, but now might be the right time in his career to do that (and he's still very young indeed). His two big successes, Porto and recently Zenit, are teams almost in the stature of Bayern in their respective leagues. He relatively failed in the Premier League due to the impatient nature of not only executive boards but also supporters. It can be argued that Chelsea and Tottenham are the more proper managerial challenges as opposed to Porto and Zenit, but Bayern aren't looking for someone to carry us to Champions League spots - we are already on top and we need someone to keep it that way. He has a vision, I think, which specifically requires exceptional and intelligent players (which is partly why he was inconsistent), but again we needn't worry about that because we already have a treasure chest worth of them. I think his biggest achievement so far is actually the record tally of points for Tottenham, as well as putting Gareth Bale on the world stage that season. Imagine what he can do with our players.
Now of course nothing's linking us with him so far, but it's looking more realistic than not because his contract expires when Pep's does. Yet still in the ideal world my preferences are firstly Pep to stay, Tuchel (only after what I saw from Dortmund, and he's not realistic anyways), Villas-Boas and then Jogi Loew.