ABSTRACT The recent thematic Assessment Report on Invasive Alien Species and their Control of the Intergovernmental Science‐Policy Platform on Biodiversity and Ecosystem Services reaffirmed biological invasions as a major threat to biodiversity. Anticipating biological invasions is crucial for avoiding their ecological and socio‐economic impacts, particularly as climate change may provide new opportunities for the establishment and spread of alien species. However, no studies have combined assessments of suitability and dispersal to evaluate the invasion by key taxonomic groups, such as mammals. Using species distribution models, we estimated the potential effect of climate change on the future distributions of 205 alien mammal species by the year 2050 under three different climatic scenarios. We used species dispersal ability to differentiate between suitable areas that may be susceptible to natural dispersal from alien ranges (Spread Potential, SP) and those that may be vulnerable to alien establishment through human‐assisted dispersal (Establishment Potential, EP) across 11 zoogeographic realms. Establishment Potential was generally boosted by climate change, showing a clear poleward shift across scenarios, whereas SP was negatively affected by climate change and limited by alien species insularity. These trends were consistent across all realms. Insular ecosystems, while being vulnerable to invasion, may act as geographical traps for alien mammals that lose climatic suitability. In addition, our analysis identified the alien species that are expected to spread or decline the most in each realm, primarily generalists with high invasive potential, as likely foci of future management efforts. In some areas, the possible reduction in suitability for alien mammals could offer opportunities for ecosystem restoration, particularly on islands. In others, increased suitability calls for adequate actions to prevent their arrival and spread. Our findings are potentially valuable in informing synergistic actions addressing both climate change and biological invasion together to safeguard native biodiversity worldwide.