Poleward transports of oceanic and atmospheric heat play an essential role in the Arctic climate system, and their variations in the future will strongly shape the climate of the Arctic. The main aim of this study is to evaluate the performance of the Coupled Model Intercomparison Project phase 6 (CMIP6) models in the historical experiment in simulating the meridional heat fluxes into the Atlantic sector of the Arctic. The secondary objective is to estimate the meridional oceanic and atmospheric heat fluxes up to the end of the 21st century using the best sub-ensembles of the CMIP6 models. According to our results, the CMIP6 models poorly reproduce the interannual variability of the heat fluxes in their historical simulations, and the multi-model ensemble mean values are systematically lower than the mean values derived from the Ocean ReAnalysis System 4 (ORAS4) and European Centre for Medium-Range Weather Forecasts Reanalysis version 5 (ERA5) reanalyses. Climate projections based on the selected CMIP6 models indicate that the future Arctic climate will be characterized by the significantly increased oceanic heat transport at the entrance to the Atlantic sector of the Arctic relative to the period 1958–2014. In contrast, the atmospheric heat and moisture transport will not have dramatic differences in the projected Arctic climate relative to the period 1958–2014. Based on the results obtained, we emphasize that any interpretation of future climate simulations should be done with caution.