In environments such as core-collapse supernovae, neutron star mergers, or the early universe, where the neutrino fluxes can be extremely high, neutrino-neutrino interactions are appreciable and contribute substantially to their flavor evolution. Such a system of interacting neutrinos can be regarded as a quantum many-body system, and prospects for nontrivial quantum correlations, i.e., entanglement, developing in a gas of interacting neutrinos have been investigated previously. In this work, we uncover an intriguing connection between the entropy of entanglement of individual neutrinos with the rest of the ensemble, and the occurrence of spectral splits in the energy spectra of these neutrinos, which develop as a result of collective neutrino oscillations. In particular, for various types of neutrino spectra, we demonstrate that the entanglement entropy is highest for the neutrinos whose locations in the energy spectrum are closest to the spectral split(s). This trend demonstrates that the quantum entanglement is strongest among the neutrinos that are close to these splits, a behavior that seems to persist even as the size of the many-body system is increased.