Abstract: The European Spallation Source (ESS), currently under construction in Sweden, will provide an intense pulsed neutrino flux allowing for high-statistics measurements of coherent elastic neutrino-nucleus scattering (CEvNS) with advanced nuclear recoil detectors. In this paper, we investigate in detail the possibility of constraining nonstandard neutrino interactions (NSIs) through such precision CEvNS measurements at the ESS, considering the different proposed detection technologies, either alone or in combination. We first study the sensitivity to neutral-current NSI parameters that each detector can reach in 3 years of data taking. We then show that operating two detectors simultaneously can significantly improve the expected sensitivity on flavor-diagonal NSI parameters. Combining the results of two detectors turns out to be even more useful when two NSI parameters are assumed to be nonvanishing at a time. In this case, suitably chosen detector combinations can reduce the degeneracies between some pairs of NSI parameters to a small region of the parameter space.