  The intra-cluster medium of several galaxy clusters hosts large-scale regions
of diffuse synchrotron radio emission, known as radio halos and relics, which
demonstrate the presence of magnetic fields and relativistic electrons in
clusters. These relativistic electrons should also emit X-rays through
inverse-Compton scattering off of cosmic microwave background photons. The
detection of such a non-thermal X-ray component, together with the radio
measurement, would permit to clearly separate the magnetic field from the
relativistic electron distribution as the inverse-Compton emission is
independent from the magnetic field in the cluster. However, non-thermal X-rays
have not been conclusively detected from any cluster of galaxies so far. In
this paper, for the first time, we model the synchrotron and inverse-Compton
emission of all clusters hosting radio halos and relics for which the spectral
index can be determined. We provide constraints on the volume-average magnetic
field by comparing with current X-ray measurements. We then estimate the
maximum volume-average magnetic field that will allow the detection of
inverse-Compton hard X-rays by the ASTRO-H satellite. We found that several
clusters are good targets for ASTRO-H to detect their inverse-Compton emission,
in particular that corresponding to radio relics, and propose a list of
promising targets for which ASTRO-H can test $\ge1$~$\mu$G magnetic fields. We
conclude that future hard X-ray observations by the already-operating NuSTAR
and the soon-to-be-launched ASTRO-H definitely have the potential to shed light
on the long-sought non-thermal hard-X-ray emission in clusters of galaxies.
