$$\require{cancel}$$
Like the magnetic field itself, the lines of magnetic vector potential form closed loops (except in the case of the infinitely long straight conducting wire, in which case they are infinitely long straight lines). That is to say $$\textbf{A}$$ has no sources or sinks, or, in other words, its divergence is everywhere zero:
$\text{div }\textbf{A}=0$