Neutron hardening and embrittlement of pressure vessel steels is due to a high density of nanometer scale features, including Cu-rich precipitates which form as a result of radiation enhanced diffusion. High-energy displacement cascades generate large numbers of both isolated point defects and clusters of vacancies and interstitials. The subsequent clustering, diffusion and ultimate annihilation of primary damage is inherently coupled with solute transport and hence, the overall chemical and microstructural evolutions under irradiation. In this work, we present atomistic simulation results, based on many-body interatomic potentials, of the migration of vacancies, solute and self-interstitial atoms (SIA) in pure Fe and binary Fe-0.9 and 1.0 at.% Cu alloys. Cu diffusion occurs by a vacancy mechanism and the calculated Cu diffusivity is in good agreement with experimental data. Strain field interactions between the oversized substitutional Cu solute atoms and SIA and SIA clusters are predominantly repulsive and result in both a decreased activation energy and diffusion pre-factor for SIA and small (N <5) SIA cluster migration, which occurs by three-dimensional motion. The Cu appears to enhance the re- orientation of the SIA clusters to different <111> directions, as well as the transition from <110> to mobile <111> configurations. The migration behavior of larger SIA clusters, which undergo only one-dimensional diffusion during molecular dynamics timescales, is largely unaffected by the Fe-Cu alloy, although SIA clusters are effectively repelled by coherent Cu precipitates.