In this paper, we report a hierarchical simulation study on the electromigration problem in Cu-CNT composite interconnects. Our work is based on the investigation of the activation energy and self-heating temperature using a multiscale electro-thermal 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 electro-thermal 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 electromigration 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 electromigration.