In spite of the known tendency for Cu to be a fast diffuser inside bulk Si, it was recently shown that it presents a clear preference for migrating towards a hydrogen terminated Si(001) surface along the dimer rows; on a surface with Bi nanolines, this will drive the Cu towards the Bi nanolines. In this paper, we present a density functional theory study of the behaviour of Cu atoms near self-assembled Bi nanolines on the hydrogen-passivated Si(001) surface, predicting that Cu will attack the nanolines and insert in the Bi-Si bonds. The migration routes from subsurface locations and surface deposition are found and the pairing tendency for Cu is examined and compared to that for Ag on Bi nanolines.