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.
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