We have studied the size confinement effect on the properties of melting-like transition of small icosahedral copper clusters using a classical molecular dynamics simulation based on a well fitted empirical potential. We investigated the caloric curves of icosahedron nanoclusters and the significant depression in the melting temperatures of the copper nanoclusters was compared with that of the bulk copper. A structural transitions from decahedral to icosahedral shapes were shown. As the cluster size increased, the melting temperature increased, and the latent heat increased but seem to be saturated. However, the specific heat was unrelated to the cluster size.