  Diffuse synchrotron radio emission has been observed in a number of cool-core
clusters on scales comparable to that of the cooling region. These radio
sources are called `mini-halos'. In order to understand their origin, which is
still unclear, joint radio and X-ray statistical studies of large cluster
samples are necessary to investigate the radio mini-halo properties and their
connection with the cluster thermodynamics. We here extend our previous
explorative study and investigate the perspectives offered by surveys in the
radio continuum with LOFAR and SKA, in particular examining the effect of the
intra-cluster magnetic field in the mini-halo region for the first time. By
considering the minimum flux detectable in radio surveys and exploiting the
$P_{radio}-L_X$ correlation observed for known mini-halos, we estimate the
detection limits achievable by future radio observational follow-up of X-ray
cluster samples, such as HIFLUGCS and eROSITA. This allows us to estimate the
maximum number of radio mini-halos that can potentially be discovered in future
surveys as a function of redshift and magnetic field strength. We show that
future radio surveys with LOFAR and SKA1 (at 140 MHz and 1.4 GHz) have the
potential to discover ~1,000-10,000 radio mini-halo candidates up to redshift
z=1. We further note that future SKA1 radio surveys at redshift z>0.6 will
allow us to distinguish between different magnetic fields in the mini-halo
region, because higher magnetic fields are expected to produce more powerful
mini-halos, thus implying a larger number of mini-halo detected at high
redshift. For example, the non-detection with SKA1 of mini-halos at z>0.6 will
suggest a low magnetic field (B < few $\mu$G). The synergy of these radio
surveys with future X-ray observations and theoretical studies is essential in
establishing the radio mini-halo physical nature. [abridged]
