Aye, they can like.
There is a standard clause in pretty much every manager's contract that stipulates they aren't allowed to bad mouth the club in public - if this leak has come direct from Rafa and the club are able to prove it, they could sack him.
Not that they would like, but it can and does happen. I'd imagine Rafa is clever enough to ensure the link has come from someone that isn't himself like, such as an associate.
He can't be sacked without full compensation just for saying he's not happy with the lack of transfer activity. That's not something that would be considered bad mouthing the club in the context of the kind of clause you're referring to.
Or am I missing the full scope of what he's meant to have said? I admit I haven't read all of the articles mentioned here.
He hasn't said anything
He hasn't been quoted as saying anything. Different from saying something to a mate who had a drink with a journo and said "Rafa thinks this and that".
That's the problem with living in a post-truth world. Thanks Rupert, ya cunt.