Using magnetic force microscopy and micromagnetic simulations, we studied the effect of Oersted magnetic fields on the chirality of transverse magnetic domain walls in Fe$_{20}$Ni$_{80}$/Ir bilayer nanostrips. Applying nanosecond current pulses with
a current density of around $2times10^{12}$ A/m$^2$, the chirality of a transverse domain wall could be switched reversibly and reproducibly. These current densities are similar to the ones used for current-induced domain wall motion, indicating that the Oersted field may stabilize the transverse wall chirality during current pulses and prevent domain wall transformations.
