The Dzyaloshinskii-Moriya interaction (DMI) is known to be responsible for multiple phenomena in magnetic materials. In the conventional description as a perturbation of the superexchange interaction by the spin-orbit coupling, the strength of the DMI is only weakly sensitive to the external fields, making its control difficult in spintronic applications. In this work, we show that an electrical modulation of the DMI may actually be possible in magnetic Weyl semimetals (WSMs). Specifically, it is theoretically illustrated that an antisymmetric indirect spin-spin interaction identified recently as an alternative mechanism for the DMI can result in the desired sensitivity to the external electric and magnetic fields via a redistribution of Weyl fermions among nodes of opposite chirality. This chiral anomaly enabled approach becomes particularly prominent in WSMs with inversion symmetry, in which the conventional DMI is not allowed. Numerical estimations suggest that moderate electric and magnetic fields of $\ensuremath{\sim}{10}^{3}--{10}^{4}\phantom{\rule{0.16em}{0ex}}\mathrm{V}/\mathrm{cm}$ and $\ensuremath{\sim}1$ T can induce a sufficiently strong change in the DMI. The impact of this externally modulated DMI on the manipulation of magnetic textures, including skyrmions in WSMs, is also discussed.