Quaternary climate has been dominated by alternating glacial and interglacial periods. While the timing and extent of past ice caps are well documented, local variations in temperature and precipitation as a response to cyclic glaciations are not resolved. Resolving these issues is necessary for understanding regional and global climate circulation. In particular, the impact of the cold high-pressure zone above the Fennoscandian ice cap on the position of the jet stream in Europe, and a possible change in the direction and the source of moisture flow are still debated. Here we reconstruct climate conditions that led to the observed ice extent in the European Alps during the last glacial maximum (LGM). Using a new inverse method to reconstruct the spatially variable position of the equilibrium line altitude (ELA), we investigate whether the Alpine LGM ice cap dominantly received precipitation in the south due to a strong southward shift of the westerlies in midlatitudes. We report inversion results that enable us to estimate the role of climate, inversion method parameters, ice dynamics, flexure and topography in modulating the inferred equilibrium line altitude. Our main finding is the increase of the position of the ELA from west to east and from north to south of the mountain range, which suggest dominant moisture delivery over the Alps during the LGM relatively similar to today.