The LEGEND (Large Enriched Germanium Experiment for Neutrinoless double-beta Decay) Collaboration will begin the construction of its initial phase, LEGEND-200, using recently re-purposed GERDA infrastructure, with a final 1000-kg installation (LEGEND-1000) in planning. A simulation study of the neutron background is underway, using a custom simulation module based on Geant4. Neutron backgrounds have a strong dependence on laboratory depth, shielding material, and cryostat design. This module has been developed to study cosmogenically-induced neutrons, as well as neutrons from (α, n) reactions. The progress and status of this work will be discussed.