A new cosmological scenario for the origin of gamma ray bursts (GRBs) is proposed. In our scenario, a highly evolved central core in the dense galactic nucleus is formed containing a subsystem of compact stellar remnants (CSRs), such as neutron stars and black holes. Those subsystems result from the dynamical evolution of dense central stellar clusters in the galactic nuclei through merging of stars, thereby forming (as has been realized by many authors) the short-living massive stars and then CSRs. We estimate the rate of random CSR collisions in the evolved galactic nuclei by taking into account, similar to Quinlan & Shapiro (1987), the dissipative encounters of CSRs, mainly due to radiative losses of gravitational waves, which results in the formation of intermediate short-living binaries, with further coalescence of the companions to produce GRBs. We also consider how the possible presence of a central supermassive black hole, formed in a highly evolved galactic nucleus, influences the CSR binary formation. This scenario does not postulate ad hoc a required number of tight binary neutron stars in the galaxies. Instead, it gives, for the most realistic parameters of the evolved nuclei, the expected rate of GRBs consistent with the observed one, thereby explaining the GRB appearance in a natural way of the dynamical evolution of galactic nuclei. In addition, this scenario provides an opportunity for a cosmological GRB recurrence, previously considered to be a distinctive feature of GRBs of a local origin only. We also discuss some other observational tests of the proposed scenario.