We consider the non-conserved dynamics of the Ising model on the two-dimensional square lattice, where each spin is influenced preferentially by its East and North neighbours. The single-spin flip rates are such that the stationary state is Gibbsian with respect to the usual ferromagnetic Ising Hamiltonian. We show the existence, in the paramagnetic phase, of a dynamical transition between two regimes of violation of the fluctuation-dissipation theorem in the nonequilibrium stationary state: a regime of weak violation where the stationary fluctuation-dissipation ratio is finite, when the asymmetry parameter is less than a threshold value, and a regime of strong violation where this ratio vanishes asymptotically above the threshold. The present study suggests that this novel kind of dynamical transition in nonequilibrium stationary states, already found for the directed Ising chain and the spherical model with asymmetric dynamics, might be quite general. In contrast with the later models, the equal-time correlation function for the two-dimensional directed Ising model depends on the asymmetry.