The aim of this experiment was to investigate whether the amount of Cu added to the diet of rats can be reduced without adversely affecting the antioxidant status of tissues and growth, and whether copper nanoparticles can be used for this purpose. For four weeks, four experimental groups of rats were fed diets with two dosages of added Cu (standard—6.5 or 3.25 mg/kg) in two forms (standard—CuCO3 or copper nanoparticles). Replacing the CuCO3 supplement with CuNPs resulted in a decreased lung weight and an increased Cu content in brain, kidney and lung, intensification of lipid peroxidation processes, and weakened antioxidant defence in the lungs and kidneys. This treatment also reduced the Cu content in heart, level of lipid oxidation in the liver and testes and improved antioxidant defence in the brain. Reducing the addition of Cu to the diet from 6.5 to 3.25 mg/kg reduced lung weight and increased lipid peroxidation in the liver, heart and lungs, and also weakened antioxidant defence in the lungs and testes. This treatment also weakened the lipid peroxidation process in the spleen, small intestine and brain and strengthened the antioxidant defence of the brain and kidneys. In conclusion, replacing CuCO3 with CuNPs and reducing the level of Cu in the diet of rats has a particularly unfavourable effect on the respiratory system, causing adverse changes in the lungs. However, these treatments have a clearly positive effect on the redox status of the liver and brain.