Abstract: Double electron capture by proton-rich nuclei is a second-order nuclear process analogous to double beta decay. Despite their similarities, the decay signature is quite different, potentially providing a new channel to measure the hypothesized neutrinoless mode of these decays. The Standard-Model-allowed two-neutrino double electron capture (2 nu EC EC) has been predicted for a number of isotopes, but only observed in Kr-78, Ba-130 and, recently, Xe-124. The sensitivity to this decay establishes a benchmark for the ultimate experimental goal, namely the potential to discover also the lepton-number-violating neutrinoless version of this process, 0 nu EC EC. Here we report on the current sensitivity of the NEXT-White detector to Xe-124 2 nu EC EC and on the extrapolation to NEXT-100. Using simulated data for the 2 nu EC EC signal and real data from NEXT-White operated with Xe-124-depleted gas as background, we define an optimal event selection that maximizes the NEXT-White sensitivity. We estimate that, for NEXT-100 operated with xenon gas isotopically enriched with 1 kg of Xe-124 and for a 5-year run, a sensitivity to the 2 nu EC EC half-life of 6 x 10(22) y (at 90% confidence level) or better can be reached.