The molecular dynamic (MD) simulations are performed to optimize the structures of the interface of carbon nanotubes and Cu nanowire. Thermal properties of their interfaces were studied on the optimized structures. The end to end interface between capped carbon nanotubes (CNTs) and uncapped CNTs were built and simulated by potential of Tersoff. The single CNT and single Cu nanowire were also built for calculation. Their thermal conductivities were calculated by a method of perturbation. The results of the simulations showed that the thermal conductivity of single CNT has a dependence on the tube length, as well as Cu nanowire. CNTs have thermal conductivity increasing with their tube length, while Cu nanowires have thermal conductivity decreasing with the length. Thermal conductivity of the interface between Cu nanowire and CNT is lower than Cu nanowire but it is higher than that between CNTs. It has very low thermal conductivity between end to end of CNTs, which is almost zero. The calculated thermal properties of the interfaces are consistent to the experiment data.