In this study, carbide-forming metallic additives (W, Mo, Cr, Ti) were introduced into the copper matrix to improve the wettability of diamond particles in the copper–diamond composites. The samples were prepared by Spark Plasma Sintering (SPS) and Hot Pressing (HP) at 920 °C. The phase composition, microstructure and thermal conductivity of the samples were investigated. The influence of the carbide-forming additive concentration, the sintering method as well as the nature of the metal introduced into the copper matrix on the thermal conductivity of copper–diamond composites was determined. Titanium ensured a more significant wettability improvement at the copper–diamond interface. This is due to its higher solubility in copper in comparison with other metals (W, Mo, Cr) and the possibility of its diffusion through the copper matrix to the diamond surface resulting in the formation of a closer contact at the copper–diamond interface.