The $Herschel$ Space telescope carried out an unprecedented survey of nearby stars for debris disks. The dust present in these debris disks scatters and polarizes stellar light in the visible part of the spectrum. We explore what can be learned with aperture polarimetry and detailed radiative transfer modelling about stellar systems with debris disks. We present a polarimetric survey, with measurements from the literature, of candidate stars observed by DEBRIS and DUNES $Herschel$ surveys. We perform a statistical analysis of the polarimetric data with the detection of far-infrared excess by $Herschel$ and $Spitzer$ with a sample of 223 stars. Monte Carlo simulations were performed to determine the effects of various model parameters on the polarization level and find the mass required for detection with current instruments. Eighteen stars were detected with a polarization $0.01 le P lesssim 0.1$ per cent and $ge3sigma_P$, but only two of them have a debris disk. No statistically significant difference is found between the different groups of stars, with, without, and unknown status for far-infrared excess, and presence of polarization. The simulations show that the integrated polarization is rather small, usually $< 0.01$ per cent for typical masses detected by their far-infrared excess for hot and most warm disks. Masses observed in cold disks can produce polarization levels above $0.01$ per cent since there is usually more dust in them than in closer disks. We list five factors which can explain the observed low-polarization detection rate. Observations with high-precision polarimeters should lead to additional constraints on models of unresolved debris disks.
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