The aim of this experiment was to test the effect of the partial or complete replacement of traditional CuCO3 in the diet of rats with copper nanoparticles (CuNPs) on the biochemical parameters, redox status, and histomorphometry of their tissues. Normotensive male Wistar–Kyoto rats (WKY) were allocated to three groups. Three analogous groups of spontaneously hypertensive rats (SHR) were also formed. The WKY and SHR rats received copper in a standard daily dose—6.5 mg/kg CuCO3 or CuNPs (100% replacement) or 3.25 mg/kg CuCO3 plus 3.25 mg/kg CuNPs (50% replacement)—for 8 weeks. Next, blood, heart, small intestine, liver, and kidney samples were collected. The activity of alanine aminotransferase, aspartate aminotransferase, creatine kinase, and gamma-glutamyl transferase and the content of creatinine and urea acid were measured in the plasma. The collected tissues were subjected to a histological evaluation, and redox status parameters (catalase and superoxide dismutase activity, malondialdehyde and glutathione content) were determined. The replacement of CuCO3 with CuNPs in the diet may exacerbate the negative changes induced by hypertension in the heart, liver, and intestines. However, it seems that it is only in the case of the liver where the observed changes may be due to an increase in oxidative reactions resulting from the inclusion of CuNPs.