  We explore the potential of current and next generation of coherent elastic
neutrino-nucleus scattering (CE$\nu$NS) experiments in probing neutrino
electromagnetic interactions. On the basis of a thorough statistical analysis,
we determine the sensitivities on each component of the Majorana neutrino
transition magnetic moment (TMM), $\left \vert \Lambda_i \right \vert$, that
follow from low-energy neutrino-nucleus experiments. We derive the sensitivity
to neutrino TMM from the first CE$\nu$NS measurement by the COHERENT
experiment, at the Spallation Neutron Source. We also present results for the
next phases of COHERENT using HPGe, LAr and NaI[Tl] detectors and for reactor
neutrino experiments such as CONUS, CONNIE, MINER, TEXONO and RED100. The role
of the CP violating phases in each case is also briefly discussed. We conclude
that future CE$\nu$NS experiments with low-threshold capabilities can improve
current TMM limits obtained from Borexino data.
