Context. Ultra-luminous X-ray sources (ULXs) are those X-ray sources located away from the centre of their host galaxy with luminosities exceeding the Eddington limit of a stellar-mass black hole ( L _X  >  10 ³⁹  erg s ⁻¹ ). Observed X-ray variability suggests that ULXs are X-ray binary systems. The discovery of X-ray pulsations in some of these objects (e.g. M82 X-2) suggests that a certain fraction of the ULX population may have a neutron star as the accretor.

Aims. We present systematic modelling of low- and intermediate-mass X-ray binaries (LMXBs and IMXBs; donor-star mass range 0.92–8.0  M _⊙ and neutron-star accretors) to explain the formation of this sub-population of ULXs.

Methods. Using MESA, we explored the allowed initial parameter space of binary systems consisting of a neutron star and a low- or intermediate-mass donor star that could explain the observed properties of ULXs. These donors are transferring mass at super-Eddington rates while the accretion is limited locally in the accretion disc by the Eddington limit. Thus, our simulations take into account beaming effects and also include stellar rotation, tides, general angular momentum losses, and a detailed and self-consistent calculation of the mass-transfer rate.

Results. Exploring the initial parameters that lead to the formation of neutron-star ULXs, we study the conditions that lead to dynamical stability of these systems, which depends strongly on the response of the donor star to mass loss. Using two values for the initial neutron star mass (1.3  M _⊙ and 2.0  M _⊙ ), we present two sets of mass-transfer calculation grids for comparison with observations of NS ULXs. We find that LMXBs/IMXBs can produce NS-ULXs with typical time-averaged isotropic-equivalent X-ray luminosities of between 10 ³⁹ and 10 ⁴¹  erg s ⁻¹ on a timescale of up to ∼1.0 Myr for the lower luminosities. Finally, we estimate their likelihood of detection, the types of white-dwarf remnants left behind by the donors, and the total amount of mass accreted by the neutron stars.

Conclusions. We show that observed super-Eddington luminosities can be achieved in LMXBs/IMXBs undergoing non-conservative mass transfer while assuming geometrical beaming. We also compare our results to the observed pulsating ULXs and infer their initial parameters. Our results suggest that a large subset of the observed pulsating ULX population can be explained by LMXBs/IMXBs in a super-Eddington mass-transfer phase.