We explore the potential of current and next generation of coherent elastic neutrino-nucleus scattering (CE$ u$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$ u$NS measurement by the COHERENT experiment, at the Spallation Neutron Source. We also present results for the next phases of COHERENT using HPGe, LAr and NaI[Tl] detectors and for reactor neutrino experiments such as CONUS, CONNIE, MINER, TEXONO and RED100. The role of the CP violating phases in each case is also briefly discussed. We conclude that future CE$ u$NS experiments with low-threshold capabilities can improve current TMM limits obtained from Borexino data.
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