Gypsum, a mineral that requires water to form, is common on the surface of Mars. Most of it originated before 3.5 Gyr when the Red Planet was more humid than now. However, occurrences of gypsum dune deposits around the North Polar Residual Cap (NPRC) seem to be surprisingly young: late Amazonian in age. This shows that liquid water was present on Mars even at times when surface conditions were as cold and dry as the present-day. A recently proposed mechanism for gypsum formation involves weathering of dust within ice (e.g., Niles, P.B., Michalski, J. [2009]. Nat. Geosci. 2, 215–220.). However, none of the previous studies have determined if this process is possible under current martian conditions. Here, we use numerical modelling of heat transfer to show that during the warmest days of the summer, solar irradiation may be sufficient to melt pure water ice located below a layer of dark dust particles (albedo ⩽ 0.13) lying on the steepest sections of the equator-facing slopes of the spiral troughs within martian NPRC. During the times of high irradiance at the north pole (every 51 ka; caused by variation of orbital and rotational parameters of Mars e.g., Laskar, J. et al. [2002]. Nature 419, 375–377.) this process could have taken place over larger parts of the spiral troughs. The existence of small amounts of liquid water close to the surface, even under current martian conditions, fulfils one of the main requirements necessary to explain the formation of the extensive gypsum deposits around the NPRC. It also changes our understanding of the degree of current geological activity on Mars and has important implications for estimating the astrobiological potential of Mars.