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 \(2\times10^{12}\) 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.