Recently we have elaborated a new cosmological model of gamma-ray burst (GRB) origin [1], which employs the dynamical evolution of central dense stellar clusters in the galactic nuclei. Those clusters inevitably contain a large fraction of compact stellar remnants (CSRs), such as neutron stars (NSs) and stellar mass black holes (BHs), and close encounters between them result in radiative captures into short-living binaries, with subsequent merging of the components thereby producing GRBs, typically at large distances from the nucleus. In the present paper, we calculate the redshift distribution of the rate of GRBs produced by close encounters of NSs in distant galactic nuclei. To this end, the following steps are undertaken: (i) we establish a connection between the parameters of the fast evolving central stellar clusters (i.e. those for which the time of dynamical evolution exceeds the age of the Universe) with masses of the forming central supermassive black holes (SMBHs) using a dynamical evolution model; (ii) we connect these masses with the inferred mass distributions of SMBHs in the galactic nuclei and the redshift distribution of quasars by assuming a certain ‘Eddington luminosity phase’ in their activity; (iii) we incorporate available observational data on the redshift distribution of quasars as well as a recently found correlation between the masses of galaxies and their central SMBHs. The resulting redshift distribution of the GRB rate, which accounts for both fast and slowly evolving galactic nuclei is consistent with that inferred from the BATSE data if the fraction of fast evolving galactic nuclei is in the range&egr;≃0.016−0.16.©1999 American Institute of Physics.