  X-ray binaries with black hole (BH) accretors and massive star donors at
short orbital periods of a few days can evolve into close binary BH systems
(BBH) that merge within the Hubble time through stable mass transfer evolution.
From observational point of view, upon the Roche-lobe overflow, such systems
will most likely appear as ultra-luminous X-ray sources (ULXs). To study this
connection, we compute the mass transfer phase in systems with BH accretors and
massive star donors ($M > 15 \,M_\odot$) at various orbital separations and
metallicities using the MESA stellar evolution code. In the case of
core-hydrogen and core-helium burning donors (cases A and C of mass transfer)
we find the typical duration of super-Eddington mass transfer of up to $10^6$
and $10^5 \, \rm yr$ , with rates of $10^{-6}$ and $10^{-5} \,M_\odot \, \rm
yr^{-1}$ , respectively. Given that roughly $0.5$ ULXs are found per unit of
star formation rate ($\,M_\odot \, \rm yr^{-1}$), and assuming that $10\%$ of
all the observed ULXs form merging BBH, we estimate the rate of BBH mergers
from stable mass transfer evolution to be at most $10 {\rm ~Gpc}^{-3} {\rm
~yr}^{-1}$.
