The pressure dependence of the x-ray magnetic circular dichroism (XMCD) has been measured in spinel ferrites (Fe₃O₄ and CoFe₂O₄). Room temperature XMCD spectra have been collected at the Fe K-edge measuring the difference in absorption between two helicity states of the photon. We measured from ambient pressure up to almost 23 GPa in the case of Fe₃O₄ and to 27 GPa in case of CoFe₂O₄. The XMCD spectra of Fe₃O₄ decrease monotonically as the pressure increases being the dichroic signal at 23 GPa half of that observed at ambient pressure. On the other hand, the dichroic signal almost disappears at 27 GPa in CoFe₂O₄. This result evidences a pressure-induced phase transition from a ferromagnetic state to a nonmagnetic state. We note that the two signals, one from the tetrahedral and the other from the octahedral Fe atoms, disappear simultaneously indicating a strong reduction of the ferromagnetic order of the two sub-lattices. In addition to this phenomenology, a clear variation of the XANES spectra at the same pressure indicates that this transition comes from a structural change. Moreover, when releasing the pressure, the dichroic signal is not recovered, even in new measurements at ambient pressure.