Calculations of laminar-turbulent transition of boundary layers are presented with low-Reynolds number versions of k-ε model of turbulence developed for the near-wall low-Reynolds-number flows. The models are tested for the transitional boundary-layers at various levels of free-stream turbulence with and without the pressure gradient. The test calculations of flows without the pressure gradient show that the model by Launder-Sharma and that by Lam-Bremhorst to some extent are able to handle the transition phenomena within a reasonable range of initial conditions of dissipation. The former model can reproduce the observed effects of level and length scale of free-stream turbulence on the transition, though the latter fails at the lower turbulence level and for the effect of length scale. It is also noted that the applicable range of the low-Reynolds-number k-ε model is limited to a relatively small pressure-gradient condition, as the model shows an opposite trend to the experimental effect of pressure gradient on the transition.