Continuous-attractor network models of grid formation posit that recurrent connectivity between grid cells controls their patterns of co-activation. Grid cells from a common module exhibit stable offsets in their periodic spatial tuning curves across environments, and this may reflect recurrent connectivity or correlated sensory inputs. Here we explore whether cell-cell relationships predicted by attractor models persist during sleep states in which spatially informative sensory inputs are absent. We recorded ensembles of grid cells in superficial layers of medial entorhinal cortex during active exploratory behaviors and overnight sleep. Per grid cell pair and collectively, and across waking, rapid eye movement sleep and non-rapid eye movement sleep, we found preserved patterns of spike-time correlations that reflected the spatial tuning offsets between these grid cells during active exploration. The preservation of cell-cell relationships across waking and sleep states was not explained by theta oscillations or activity in hippocampal subregion CA1. These results indicate that recurrent connectivity within the grid cell network drives grid cell activity across behavioral states.