The effect of quantum well states (QWS) formed in a Cu overlayer on the magnetic anisotropy of Fe and of Co films underneath is investigated at low temperatures. An oscillatory magnetic anisotropy is observed below 50 K, with a period of L_Cu = 5.8 ML, and it is the same for Cu/Fe and Cu/Co. We attribute this effect to QWS from sp electrons in Cu which hybridize to d electrons at the Fermi level in Fe (and Co) and thus modify the magnetic anisotropy at low temperatures. This is why the effect was not so pronounced in the experiments reported 15 years ago showing the low-amplitude oscillations detected at 170 K [Ch. Würsch et al., Nature (London) 389, 937 (1997)].