What is a thing that is homomorphized? This is a purely syntax question. If $A$ is a structure and $\pi$ is a homomorphism sending $A$ to something.
We say that $\pi$ is a homomorphism of $A$.
Similarly we might want to say $A$ is a ____ of $\pi$.
What is that word? I found myself writing "$A$ is a structure for which $\pi$ is a homomorphism" but that really was verbose and convoluted.
What I had considered:
$A$ is a domain, but this doesn't carry the importance that $\pi$ is a homomorphism of $A$, it only says that $\pi$ is a function which maps $A$ to something.
 A: I'm not aware of any context where we have a fixed map $\pi$ but consider different structures on the domain of $\pi$ with respect to which $\pi$ remains a homomorphism, and this is the only sort of context I can think of where "$A$ is the domain of $\pi$" would seem inappropriate to me. So I don't think such a term exists.
(Contrast this with "$\pi$ is a homomorphism from $A$" - this language is useful since we do often consider multiple homomorphisms out of the same structure.)

EDIT: you clarify that this is in fact the context you're in. I stand by my claim that this context is rare enough that there probably isn't a standard term in the literature. That said, the term which fits the bill most cleanly for me is "compatible," and I would suggest the phrase "$A$ is a compatible domain for $\pi$," or - if you only ever talk about varying domains and not codomains - "$A$ is compatible with $\pi$." But of course unless I'm wrong and there is after all a standard term this should be defined explicitly in your text.
Alternatively, a phrase like "$A$ is a domain object for $\pi$" has a categorical flavor which may be desired (this is motivated by Ty Jensen's comment above). Personally I think this sort of thing is clunkier, but I could see it being preferred.
