  We explored a suite of high-resolution cosmological simulations from the
First Billion Years Project (FiBY) at $z \geq 6$. All substructures within the
simulations have been identified with the SUBFIND algorithm. From our analysis,
two distinct groups of objects emerge. We hypothesise that the substructures in
the first group, which appear to have a high baryon fraction ($f_{\rm b} \geq
0.95$), are possible infant GC candidates. Objects belonging to the second
group have a high stellar fraction ($f_{\rm star} \geq 0.95$) and show a
potential resemblance to infant ultra-faint dwarf galaxies. The high baryon
fraction objects identified in this study are characterised by a stellar
content similar to the one observed in present-day GCs, but they still contain
a high gas fraction ($f_{\rm gas} \sim 0.95$) and a relatively low amount of
dark matter. They are compact, dense systems. Their sizes are consistent with
recent estimates based on the first observations of possible proto-GCs at high
redshifts. These types of infant GC candidates appear to be more massive and
more abundant in massive host galaxies, indicating that the assembly of
galaxies via mergers may play an important role in building several GC-host
scaling relations. Specifically, we express the relation between the mass of
the most massive infant GC and its host stellar mass as $\log(M_{\rm cl}) =
(0.31\pm0.15)\log(M_{\rm *,gal} + (4.17\pm1.06)$. We also report a new relation
between the most massive infant GC and the parent specific star formation rate
of the form $\log(M_{\rm cl}) = (0.85\pm0.30)\log(sSFR) + \alpha$ that
describes the data at both low and high redshift. Finally, we assess the
present-day GC mass (GC number) -- halo mass relation offers a satisfactory
description of the behaviour of our infant GC candidates at high redshift,
suggesting that such a relation may be set at formation.
