Why can't you, in addition to buying and selling players, include a transfer window by making offers to other players?
For example, I'm interested in a player from team X, and I make him an offer.

Would be great indeed. 

In some games, you can set an asking price for your players so that anyone can bid that amount and start an auction at any moment, in addition to putting players on auction. Kinda like a release clause

That could be a nice addition, instead of having to contact the user and ask for the price