Given $n$, the set of  integers $m$ coprime to $n$ has nonzero asymptotic density, thus so does the set $mn$.
But then $\phi(n)/n > \phi(mn)/mn$ , so the $\liminf_n\phi(n)/n$ will remain the same off of a set of zero asymptotic density. So there is no such $f(\cdot)$ for asymptotic density. (I assume you mean + epsilon in your formulation.)