This study is focused on the design of new Al–Cu–Mn alloys with enhanced strength and heat resistance achieved without high-temperature exposure. There are seven alloys considered with permanent Mn content of 2% and variable Cu content of 0–4%. It is found that ~2% Mn is almost completely dissolved in the solid solution after casting, while the copper is distributed between the solid solution and Al2Cu eutectic constituents. It is experimentally established that, when the copper content is 2 and 3%, the solid solution of aluminum contains approximately the same amount of copper, up to 1.5%. The deformation plasticity of experimental alloys in the cold rolling mode with the reduction rate of 80 and 95% is investigated. It is shown that no preliminary treatment is required for alloys containing up to 3% Cu, as they exhibit high processability at rolling. The effect of heat treatment in the annealing temperature range of 200–600°C on the structural and phase parameters of the alloys is studied. Alloy hardening in the process of multistage annealing by means of hardness measurement is estimated. The resulting data allow us to determine the influence of copper and analyze the thermal stability of the model alloys. Tensile tests of cold-rolled sheets with a thickness of 0.5 mm and a reduction level of 95% of alloys containing 2 and 3% copper show high values. In particular, the alloy containing 3% Cu has a strength performance compatible with that of 1201 alloy in T6 condition.