Assessing invasion risk by alien organisms implies evaluating the likelihood of successful establishment and spread once they are accidentally or deliberately introduced. The importance of implementing accurate risk–assessment procedures is further stressed by the evidence that ongoing climate change can promote invasion processes, from initial introduction through establishment and spread. Although squirrels are considered powerful invaders with well–documented negative impacts on biodiversity and human activities, there is a noticeable gap of comprehensive investigations specifically focused on this group, especially as far as the effect of climate change on worldwide invasion risk is concerned. In this study we predicted current and 2070 potential distributions of eight squirrel species with a SDM–based framework, also detecting current potential hotspots of invasion and evaluating how these could be modified by climate change in 2070. SDM predicted the eight species to potentially occur in large areas worldwide (12.12% to 39.23% of the mainland), with 70 to 129 countries vulnerable to a potential invasion. Model projections over the 2070 climate change scenarios predicted five to seven squirrel species undergoing an increase in their future potential distribution in non–native ranges. Current hotspots of potential invasion were predicted to occur in southeastern Asia, northeastern Australia, tropical Africa and South America, as well as Central and North America. Projections to 2070 showed most of the hotspots of invasion to remain substantially stable in terms of number of potential invasive species, regardless of the scenario. The most relevant increase/reduction in extent of species distribution and in the number of potential invasive species in invasiveness hotspots were predicted for 2070 under the most severe scenarios. We emphasize a strong species–specific response to climate change, which could also affect invasive species by making them less competitive, therefore potentially leading to a retreat from the invaded ranges.