Recent studies note a significant increase in high-pressure blocking over the Greenland region (Greenland Blocking Index, GBI) in summer since the 1990s. Such a general circulation change, indicated by a negative trend in the North Atlantic Oscillation (NAO) index, is generally highlighted as a major driver of recent surface melt records observed on the Greenland Ice Sheet (GrIS). Here we compare reanalysis-based GBI records with those from the Coupled Model Intercomparison Project 5 (CMIP5) suite of global climate models over 1950–2100. We find that the recent summer GBI increase lies well outside the range of modelled past reconstructions and future GBI projections (RCP4.5 and RCP8.5). The models consistently project a future decrease in GBI (linked to an increase in NAO), which highlights a likely key deficiency of current climate models if the recently observed circulation changes continue to persist. Given well-established connections between atmospheric pressure over the Greenland region and air temperature and precipitation extremes downstream, e.g. over northwest Europe, this brings into question the accuracy of simulated North Atlantic jet stream changes and resulting climatological anomalies over densely populated regions of northern Europe as well as of future projections of GrIS mass balance produced using global and regional climate models.