Yes, there is a completely robust mechanism for introducing self-reference into mathematics. Any sufficiently robust mathematical language will admit the possibility of self-reference, as shown by the Gödel-Carnap fixed-point lemma. In particular, there is no problem at all with having a sentence refer to itself or to the prior sentences in some construction or the later sentences.

For example, one can consider the provability version of the Visser-Yablo paradox, a Gödelized infinite liar, which produces a uniformly expressible sequence of sentences $\sigma(k)$ in the language of arithmetic, each of which provably expresses the non-provability of all the later sentences
$$\text{PA}\vdash\quad\forall k\left[\sigma(k)\leftrightarrow\forall n{>}k\ \left(\strut\neg\text{Pr}_{\text{PA}}(\ulcorner\sigma(\bar n)\urcorner)\right)\right].$$
I consider these sentences in my elementary essay on the Infinite Liars. So, the main lesson is that self-reference is no problem. With quite flexible methods, one can find a solution to nearly any self-referential expressible condition.

Let me emphasize, however, that self-referentiality means that sentences can refer to any expressible properties of themeselves and the theory in which they are described. This would include any kind of syntactic feature of the sentences, such as provability, consistency, satisfiability, but also $\omega$-realizability. In set theory one can form self-referential theories that refer to the $\omega$-realizability of themselves or other parts of the sentences.

In particular, we can form form a formula $\varphi(x)$ such that ZFC proves that for every natural number $k$, $\varphi(k)$ asserts that the theory arising from $\varphi(n)$ for $n\neq k$ is true in some $\omega$-standard model. The existence of such a formula is an immediate instance of the Gödel-Carnap fixed point lemma.

If we would mean "an $\omega$-model of ZFC" for the sentences, then it is consistent with ZFC that they are false, since it is consistent with ZFC that there is no $\omega$-model of ZFC at all. I'd have to think more carefully to determine if they can be true, or what happens if one wants a model just of this theory and not also the ambient theory, such as ZFC here.

Meanwhile, let me emphasize that we cannot form self-referential sentences that refer to the *truth* or *falsity* of themselves or other parts of the theory. In light of Tarski's theorem on the nondefinability of truth, truth and falsity are not uniformly expressible features of sentences.