Context. It is now established that magnetic fields are present in the intra-cluster medium (ICM) of galaxy clusters, as revealed by observations of radio halos and radio relics and from the study of the Faraday rotation measures of sources located either behind or within clusters. Deep radio polarization observations of clusters have been performed in the last years, and the properties of the ICM magnetic field have been constrained in a small number of well-studied objects. Aims: The aim of this work is to investigate the average properties of the ICM magnetic fields, and to search for possible correlations with the ICM thermal properties and cluster radio emission. Methods: We have selected a sample of 39 massive galaxy clusters from the HIghest X-ray FLUx Galaxy Cluster Sample, and used Northern VLA Sky Survey data to analyze the fractional polarization of radio sources out to 10 core radii from the cluster centers. We have investigated how different magnetic field strengths affect the observed polarized emission of sources lying at different projected distances from the cluster center. In addition, statistical tests are performed to investigate the fractional polarization trends in clusters with different thermal and non-thermal properties. Results: We find a trend of the fractional polarization with the cluster impact parameter, with fractional polarization increasing at the cluster periphery and decreasing toward the cluster center. Such trend can be reproduced by a magnetic field model with central value of few muG. The logrank statistical test indicates that there are no differences in the depolarization trend observed in cluster with and without radio halo, while the same test indicates significant differences when the depolarization trend of sources in clusters with and without cool core are compared. The comparison between clusters with high and low temperatures does not yields significant differences. Although the role of the gas density should be better accounted for, these results give important indications for models that require a role of the ICM magnetic field to explain the presence of cool core and radio halos in galaxy clusters.