We present a detailed study regarding the bandgap dependence on diameter and composition of spherical Ge-rich GexSi1-x nanocrystals (NCs). For this, we conducted a series of atomistic density functional theory (DFT) calculations on H-passivated NCs of Ge-rich GeSi random alloys, with Ge atomic concentration varied from 50 to 100% and diameters ranging from 1 to 4 nm. As a result of the dominant confinement effect in the DFT computations, a composition invariance of the line shape of the bandgap diameter dependence was found for the entire computation range, the curves being shifted for different Ge concentrations by ΔE(eV) = 0.651(1 - x). The shape of the dependence of NCs bandgap on the diameter is well described by a power function 4.58/d1.25 for 2-4 nm diameter range, while for smaller diameters, there is a tendency to limit the bandgap to a finite value. By H-passivation of the NC surface, the effect of surface states near the band edges is excluded aiming to accurately determine the NC bandgap. The number of H atoms necessary to fully passivate the spherical GexSi1-x NC surface reaches the total number atoms of the Ge + Si core for smallest NCs and still remains about 25% from total number of atoms for bigger NC diameters of 4 nm. The findings are in line with existing theoretical and experimental published data on pure Ge NCs and allow the evaluation of the GeSi NCs behavior required by desired optical sensor applications for which there is a lack of DFT simulation data in literature.