Setting the formation of globular clusters (GCs) within a cosmological context and characterising the properties of proto-GCs at high redshift is currently a major challenge. In this work, we address that challenge by exploring a suit of high-resolution cosmological simulations from the First Billion Years (FiBY) project z at ⩾6 to investigate theoretical scenarios concerning the formation of old, low-mass stellar systems with a particular focus on GCs. Two distinct groups of objects are identified in the simulations. The first group of objects, with a high baryon fraction, we associate with proto-GCs. The second group, that exhibit a high stellar fraction, could be forming ultra-faint dwarf galaxies (UFDs). The objects with high baryon fraction are promising proto-GC candidates because they have little to no dark matter (DM), have number densities consistent with predictions from the literature, are very compact and have a high stellar density. We fit and also assess the redshift-zero globular system mass - halo mass relation and find it provides a reasonable fit to our proto-GC objects, indicating that this relation is likely set at formation.