Thyroid cancer is the most common endocrine cancer. There is no systematic screening for such cancer, and the current challenge is to find potential biomarkers to facilitate an early diagnosis. Copper (Cu) and zinc (Zn) are essential micronutrients involved in the proper functioning of the thyroid gland, and changes in their concentrations have been observed in the development of cancer. Previous studies have highlighted the potential 65Cu/63Cu ratio (δ65Cu) to be a cancer biomarker. This study tests its sensitivity on plasma samples (n = 46) of Algerian patients with papillary thyroid carcinoma and a set of corresponding biopsies (n = 11). The δ65Cu ratio in blood and tumor samples was determined using multi collector inductively coupled plasma-mass spectrometry (MC-ICP-MS), and their corresponding Cu and Zn plasma total concentrations using total reflection X-ray fluorescence (TXRF). Plasma concentrations of Cu were significantly higher (1346.1 ± 328.3 vs. 1060.5 ± 216.1 μg/L, p < 0.0001), and Zn significantly lower (942.1 ± 205.2 vs. 1027.9 ± 151.4 μg/L, p < 0.05) in thyroid cancer patients as compared to healthy controls (n = 50). Accordingly, the Cu/Zn ratio was significantly different between patients and controls (1.5 ± 0.4 vs. 1.0 ± 0.3, p < 0.0001). Furthermore, the δ65Cu plasma levels of patients were significantly lower than healthy controls (p < 0.0001), whereas thyroid tumor tissues presented high δ65Cu values. These results support the hypothesis that Cu isotopes and plasma trace elements may serve as suitable biomarkers of thyroid cancer diagnosis.