Carbon nanotubes (CNTs) are a promising candidate to replace copper interconnects. An ab initio study is presented on the conductance of a closed-packed bundle of very narrow metallic (4,0) CNTs, which is vertically placed on a Cu (100) surface. The intertube interactions have no significant impact on the conductance. The conductance is highly dependent on the exact geometry of the interface, which is varying between 0.6 and 1.8 conductance quanta, while the theoretical maximum of the CNT is three conductance quanta. The wave-function interference can lead to conductance suppression when the packing is too high. Both features are explained by using an orbital picture.