There are two hard problems of omniscience. One is metaphysical—how can a God who has aseity and is simple know contingent facts. The other is logical—how can one formulate omniscience in a way that avoids the paradoxes of truth. The paradoxes of truth are going to show up as soon as we have quantification over propositions and a predicate coextensive with truth. Every orthodox account of omniscience gives a predicate coextensive with truth: "is believed by God". The standard formulation of omniscience is that God knows all and only tue propositions. That gives us quantification over propositions, and so it seems we have everything needed for paradox.
One might say that this is a paradox everyone faces. Well, but not quite—only those who have a truth predicate face it.
Here is a solution. Don't quantify over propositions. Instead, say that just as logic allows one to infer "——" from "—— and ****", so too logic allows one to infer "——" from "x believes —— and omniscient(x)" and "x knows ——" from "——" and "omniscient(x)". This rule of inference no more gives rise to paradox than the rules of inference of first order logic.
Of course this does nothing to help with the metaphysical problem (for that, see my piece in the first Oxford Studies in Philosophy of Religion).