In this paper we study under which conditions the oscillating Urca process can be relevant for the evolution of mass-accreting white dwarfs. We have found that, if the oscillation amplitude is large enough, which corresponds to high accretion rates, it can trigger the ignition of carbon at a lower density than presently thought, at rho = 1.17 10(9) g cm-3, thus producing less neutronized chemical species than common models of type la supernovae. If the oscillation amplitude is small, which corresponds to low accretion rates, carbon can be ignited non-explosively and induce the non-explosive collapse of the white dwarf to a neutron star. We also discuss how the oscillation can appear.