We aimed to test the hypothesis of bioclimatic niche conservation in an alien species. The study location was the Northern Hemisphere temperate and boreal zones. We studied tree squirrels of the genera Sciurus and Tamiasciurus, especially S. carolinensis (Eastern Grey Squirrel). Climatic response surface species distribution models (SDMs) are fitted using a limited set of biologically-relevant bioclimatic variables. Two models are fitted for Sciurus carolinensis: one for its North American distribution alone, and the second for this plus its naturalised European distribution. Each model is used to simulate the species’ potential global distribution. Models are fitted also for S. vulgaris (Eurasian Red Squirrel), 12 further North American, and two further Eurasian tree squirrels. The realised bioclimatic niches of S. carolinensis, two species with which its North American distribution overlaps, and S. vulgaris are compared graphically. All the SDMs fitted gave excellent goodness-of-fit values. The two SDMs fitted for S. carolinensis simulate markedly different potential distributions, notably in Europe. The model fitted to the North American distribution alone simulates high suitability from France eastwards to Ukraine and onwards to Central Asia, but absence from the British Isles. That fitted including the naturalised European distribution, in contrast, simulates high suitability in the British Isles and only sparse areas of low suitability elsewhere across Europe from Spain to Ukraine and Russia. The latter model also simulates areas of high suitability in Pacific North-west North America and Chile. Graphical exploration of the realised bioclimatic niche of S. carolinensis showed the thermal niche occupied in Europe to have no overlap with that occupied in North America. The bioclimatic realised niche of S. carolinensis in Europe was not predicted by an SDM fitted to its North American distribution. Such lack of niche conservatism has important implications for the use of SDMs in screening potential alien introductions and in planning conservation translocations.