That's quite a theory, but we know that there is more than £100m in the bank, and it would be suicidal if the owners were prepared to (hypothetically) lose out on the Premier League and the Champions League to save £12m. It's possible, I suppose, that they may not care about the extra revenue that is earned from winning major trophies, or even that they believe that we can win them without further investment. But you would have to think that if the manager wanted a player for anything under £20m, the owners would consider that to be a sign that the manager believes that it is necessary, and that they would make the money available.
As it is, Werder have been very clear that they would rather keep the player and lose him on a free if they can't attract a high enough fee to allow them to replace him. The £12m figure has no real basis in fact, as far as I am aware. But the difference between £12m and, say, £16m is not particularly important, anyway.
And there is also no evidence to suggest that we are even interested in Ozil and plenty of reason to believe that we would need to sell players to make his signing worthwhile even if we were. With five strikers at the club and two very good wingers who have earned their place in the team, it isn't immediately obvious that he is either required, or that he would replace any of the players who are currently deserving of their position in the team.
The fact that some people are trying to shoe-horn him in to a central midfield position suggests that the desire to sign him has completely over-ridden all other considerations. And that's just not a particularly good reason to sign anyone, in my opinion, no matter how talented they are.