As far as I understand Arsenal is a public limited company - I will admit that I am not very knowledgeable about this financial talk.
Galatasaray is in a similar position, however, they need to declare all of their signings and the signings' total income when the transfer is finalized. Further, they have to declare when transfer talks are in play to not be charged, AFAIK, due to being a public company in the Turkish stock market.
We see a similar situation with Barcelona FC and the signing of Neymar.
So how can Manchester City and Arsenal manage to hide the exchange of money for a transfer? Shouldn't that be public knowledge?