  Although ultra-luminous X-ray sources (ULX) are important for astrophysics
due to their extreme apparent super-Eddington luminosities, their nature is
still poorly known. Theoretical and observational studies suggest that ULXs
could be a diversified group of objects composed of low-mass X-ray binaries,
high-mass X-ray binaries and marginally also systems containing
intermediate-mass black holes, which is supported by their presence in a
variety of environments. Observational data on the ULX donors could
significantly boost our understanding of these systems, but only a few were
detected. There are several candidates, mostly red supergiants (RSGs), but
surveys are typically biased toward luminous near-infrared objects.
Nevertheless, it is worth exploring if RSGs can be members of ULX binaries. In
such systems matter accreted onto the compact body would have to be provided by
the stellar wind of the companion, since a Roche-lobe overflow could be
unstable for relevant mass-ratios. Here we present a comprehensive study of the
evolution and population of wind-fed ULXs and provide a theoretical support for
the link between RSGs and ULXs. Our estimated upper limit on contribution of
wind-fed ULX to the overall ULX population is $\sim75$--$96\%$ for young
($<100$ Myr) star forming environments, $\sim 49$--$87\%$ for prolonged
constant star formation (e.g., disk of Milky Way), and $\lesssim1\%$ for
environments in which star formation ceased long time ($>2$ Gyr) ago. We show
also that some wind-fed ULXs (up to $6\%$) may evolve into merging double
compact objects (DCOs), but typical systems are not viable progenitors of such
binaries because of their large separations. We demonstrate that, the exclusion
of wind-fed ULXs from population studies of ULXs, might have lead to
systematical errors in their conclusions.
