We observed the hot Jupiter HAT-P-32b (also known as HAT-P-32Ab) to determine its optical transmission spectrum by measuring the wavelength-dependent, planet-to-star radius ratios in the region between 518-918 nm. We used the OSIRIS instrument at the Gran Telescopio CANARIAS (GTC) in long-slit spectroscopy mode, placing HAT-P-32 and a reference star in the same slit and obtaining a time series of spectra covering two transit events. Using the best quality data set, we were able to yield 20 narrowband transit light curves, with each passband spanning a 20 nm wide interval. After removal of all systematic noise signals and light curve modeling, the uncertainties for the resulting radius ratios lie between 337 and 972 ppm. The radius ratios show little variation with wavelength, suggesting a high altitude cloud layer masking any atmospheric features. Alternatively, a strong depletion in alkali metals or a much smaller than expected planetary atmospheric scale height could be responsible for the lack of atmospheric features. Our result of a flat transmission spectrum is consistent with a previous ground-based study of the optical spectrum of this planet. This agreement between independent results demonstrates that ground-based measurements of exoplanet atmospheres can give reliable and reproducible results despite the fact that the data often is heavily affected by systematic noise as long as the noise source is well understood and properly corrected. We also extract an optical spectrum of the M-dwarf companion HAT-P-32B. Using PHOENIX stellar atmosphere models we determine an effective temperature of Teff = 3187+60-71 K, which is slightly colder than previous studies relying only on broadband infrared data. The 20 narrowband and white light curves are only available at the CDS via anonymous ftp to http://cdsarc.u-strasbg.fr (http://130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/594/A65