We present the results of hydrodynamical simulations of the orbital evolution of planets undergoing runaway gas accretion in radiative discs. We consider accreting disc models with constant mass flux through the disc, and where radiative cooling balances the effect of viscous heating and stellar irradiation. We assume that 20-30 $M_\oplus$ giant planet cores are formed in the region where viscous heating dominates and migrate outward under the action of a strong corotation torque. In the case where gas accretion is neglected, we find evidence for strong dynamical torques in accreting discs with accretion rates ${\dot M}\gtrsim 7\times 10^{-8} \;M_\odot/yr$. Their main effect is to increase outward migration rates by a factor of $\sim 2$ typically. In the presence of gas accretion, however, runaway outward migration is observed with the planet passing through the zero-torque radius and the transition between the viscous heating and stellar heating dominated regimes. The ability for an accreting planet to enter a fast migration regime is found to depend strongly on the planet growth rate, but can occur for values of the mass flux through the disc of ${\dot M}\gtrsim 5\times 10^{-8} \;M_\odot/yr$. We find that an episode of runaway outward migration can cause an accreting planet formed in the 5-10 AU region to temporarily orbit at star-planet separations as large as $\sim$60-70 AU. However, increase in the amplitude of the Lindblad torque associated with planet growth plus change in the streamline topology near the planet systematically cause the direction of migration to be reversed. Our results indicate that a planet can reach large orbital distances under the combined effect of dynamical torques and gas accretion, but an alternative mechanism is required to explain the presence of massive planets on wide orbits.