Thermal interface materials (TIMs) are essential for managing heat in modern electronics, and nanocomposite TIMs can offer critical improvements. Here, we demonstrate thermally conductive, mechanically compliant TIMs based on dense, vertically aligned copper nanowires (CuNWs) embedded into polymer matrices. We evaluate the thermal and mechanical characteristics of 20-25% dense CuNW arrays with and without polydimethylsiloxane infiltration. The thermal resistance achieved is below 5 mm K W, over an order of magnitude lower than commercial heat sink compounds. Nanoindentation reveals that the nonlinear deformation mechanics of this TIM are influenced by both the CuNW morphology and the polymer matrix. We also implement a flip-chip bonding protocol to directly attach CuNW composites to copper surfaces, as required in many thermal architectures. Thus, we demonstrate a rational design strategy for nanocomposite TIMs that simultaneously retain the high thermal conductivity of aligned CuNWs and the mechanical compliance of a polymer.