  During their evolution, star clusters undergo mass segregation, by which the
orbits of the most massive stars shrink, while the lighter stars move outwards
from the cluster centre. In this context, recent observations and dynamical
modelling of several galactic and extra-galactic globular clusters (GCs)
suggest that most of them show, close to their centre, an overabundance of mass
whose nature is still matter of debate. For instance, many works show that
orbitally segregated stars may collide with each other in a runaway fashion,
leading to the formation of a very massive star or an intermediate mass black
hole (IMBH) with a mass comparable to the observed mass excess. On the other
hand, segregated stars can form a dense system if the IMBH formation fails. In
this paper we study the early formation phase of a dense, massive sub-system
(MSS) in several GCs models using a recently developed semi-analytical
treatment of the mass segregation process. In order to investigate how the MSS
properties depend on the host cluster properties, we varied initial mass
function (IMF), total mass, spatial distribution and metallicity of our models.
Our results show how the IMF contributes to determine the final mass of the
MSS, while the metallicity and the spatial distribution play a minor role. The
method presented in this paper allowed us to provide scaling relations that
connect the MSS mass and the host cluster mass in agreement with the observed
correlation. In order to follow the early formation stage of the MSSs and
improve our statistical results, we performed several $N$-body simulations of
stellar clusters with masses between $10^3$ and $2\times 10^5$ solar masses.
