The Dzyaloshinskii-Moriya (DM) interaction is an antisymmetric exchange interaction that is responsible for the emergence of chiral magnetism. The origin of the DM interaction, however, remains to be identified albeit the large number of studies reported on related effects. It has been recently suggested that the DM interaction is equivalent to an equilibrium spin current density originating from spin-orbit coupling, an effect referred to as the spin Doppler effect. The model predicts that the DM interaction can be controlled by spin current injected externally. Here we show that the DM exchange constant (D) in W/CoFeB-based heterostructures can be modulated with external current passed along the film plane. At higher current, D decreases with increasing current, which we infer is partly due to the adiabatic spin transfer torque. At lower current, D increases linearly with current regardless of the polarity of current flow. The rate of increase in D with the current density agrees with that predicted by the model based on the spin Doppler effect. These results imply that the DM interaction at the heavy-metal-ferromagnetic-metal interface partly originates from an equilibrium interface spin (polarized) current which can be modulated externally.