  We have studied the interfacial Dzyaloshinskii-Moriya interaction (DMI) in a
series of Pt/Co/MOx (M=Al, Gd) trilayers in which the degree of oxidation of
the top Co interface is varied. To access to reliable values of the DMI
strength, we have used a method based on the measurement of the saturation
velocity of field driven chiral N\'eel domain walls. We show that the effective
DMI strength in the Pt/Co/MOx trilayers varies with the oxidation degree of the
Co/MOx interface. This strongly suggests that the Co/MOx interface gives a
distinct contribution to the total DMI, adding to that of the Pt/Co interface.
The DMI presents a maximum for the oxygen coverage maximizing also the
interface magnetic anisotropy energy Ks. This calls for common microscopic
origins for the contributions of the Co/MOx interface to DMI and Ks.
