In this paper, we report a hierarchical simulation study of the electromigration (EM) problem in Cu-carbon nanotube (CNT) composite interconnects. This paper is based on the investigation of the activation energy and self-heating temperature using a multiscale electrothermal simulation framework. We first investigate the electrical and thermal properties of Cu-CNT composites, including contact resistances, using the density functional theory and reactive force field approaches, respectively. The corresponding results are employed in macroscopic electrothermal simulations taking into account the self-heating phenomenon. Our simulations show that although Cu atoms have similar activation energies in both bulk Cu and Cu-CNT composites, Cu-CNT composite interconnects are more resistant to EM thanks to the large Lorenz number of the CNTs. Moreover, we found that a large and homogenous conductivity along the transport direction in interconnects is one of the most important design rules to minimize the EM.