A new cosmological scenario for the origin of gamma-ray bursts (GRBs) is pr
oposed. In our scenario, a highly evolved central core in the dense galacti
c nucleus is formed, containing a subsystem of compact stellar remnants (CS
Rs), such as neutron stars and black holes. Those subsystems result from th
e dynamical evolution of dense central stellar clusters in the galactic nuc
lei through merging of stars, thereby forming (as has been realized by many
authors) the short-lived massive stars and then CSRs. We estimate the rate
of random CSR collisions in the evolved galactic nuclei by taking into acc
ount, in a procedure similar to that of Quinlan & Shapiro, the dissipative
encounters of CSRs, mainly due to radiative losses of gravitational waves,
which result in the formation of intermediate short-lived binaries, with fu
rther coalescence of the companions to produce GRBs. We also consider how t
he possible presence of a central supermassive black hole, formed in a high
ly evolved galactic nucleus, influences the CSR binary formation. This scen
ario does not postulate ad hoc a required number of tight binary neutron st
ars in the galaxies. Instead, it gives, for the most realistic parameters o
f the evolved nuclei, the expected rate of GRBs consistent with the observe
d one, thereby explaining the GRB appearance as a natural part of the dynam
ical evolution of galactic nuclei. In addition, this scenario provides an o
pportunity for a cosmological GRB recurrence, previously considered to be a
distinctive feature of GRBs of a local origin only. We also discuss some o
ther observational tests of the proposed scenario.