  We investigate how the proximity of supergiant donor stars to the
Eddington-limit, and their advanced evolutionary stage, may influence the
evolution of massive and ultra-luminous X-ray binaries with supergiant donor
stars (SGXBs and ULXs). We construct models of massive stars with different
internal hydrogen/helium gradients and different hydrogen-rich envelope masses,
and expose them to slow mass loss to probe the response of the stellar radius.
In addition, we compute the corresponding Roche-lobe overflow mass-transfer
evolution with our detailed binary stellar evolution code, approximating the
compact objects as point masses. We find that a hydrogen/helium gradient in the
layers beneath the surface, as it is likely present in the well-studied donor
stars of observed SGBXs, can enable nuclear timescale mass-transfer in SGXBs
with a BH or a NS accretor, even for mass ratios in excess of 20. In our binary
evolution models, the donor stars rapidly decrease their thermal equilibrium
radius and can therefore cope with the inevitably strong orbital contraction
imposed by the high mass ratio. Our results open a new perspective for
understanding the large number of Galactic SGXBs, and their almost complete
absence in the SMC. They may also offer a way to obtain more ULX systems, to
find nuclear timescale mass-transfer in ULX systems even with neutron star
accretors, and shed new light on the origin of the strong B-field in these
neutron stars.
