ABSTRACT We report the formation of bound star clusters in a sample of high-resolution cosmological zoom-in simulations of z ≥ 5 galaxies from the Feedback In Realistic Environments project. We find that bound clusters preferentially form in high-pressure clouds with gas surface densities over $10^4\, \mathrm{ M}_{⊙ }\, {\rm pc}^{-2}$, where the cloud-scale star formation efficiency is near unity and young stars born in these regions are gravitationally bound at birth. These high-pressure clouds are compressed by feedback-driven winds and/or collisions of smaller clouds/gas streams in highly gas-rich, turbulent environments. The newly formed clusters follow a power-law mass function of dN/dM ∼ M−2. The cluster formation efficiency is similar across galaxies with stellar masses of ∼107–$10^{10}\, \mathrm{ M}_{⊙ }$ at z ≥ 5. The age spread of cluster stars is typically a few Myr and increases with cluster mass. The metallicity dispersion of cluster members is ∼0.08 dex in $\rm [Z/H]$ and does not depend on cluster mass significantly. Our findings support the scenario that present-day old globular clusters (GCs) were formed during relatively normal star formation in high-redshift galaxies. Simulations with a stricter/looser star formation model form a factor of a few more/fewer bound clusters per stellar mass formed, while the shape of the mass function is unchanged. Simulations with a lower local star formation efficiency form more stars in bound clusters. The simulated clusters are larger than observed GCs due to finite resolution. Our simulations are among the first cosmological simulations that form bound clusters self-consistently in a wide range of high-redshift galaxies.