The following statement has been largely improved from my original post thanks to discussions with @DmitriPanov ad well as the comment from @Wojowu.

Let $f \colon \mathbb{S}^3 \to \mathbb{R}^2$ be real analytic. Let $C$ be a closed space curve in $\mathbb{S}^3$, which I might need to assume to be unknot. If $f(C) \subset \mathbb{R}^2$ has a non-zero winding number around $0$, then $f^{-1}(0)$ contains a space curve in $\mathbb{S}^3$ that is linked to $C$.

I consider this as a generalization of Kronecker's existence theorem. I sketched a plausible argument in my comment to the answer of @DmitriPanov, which I did not check with great care. The statement and argument is obviously generalizable to other dimensions, using topological degree.

I believe that this result is not new. So where do I find a reference for this and its higher dimensional versions?