The consumption of marine fishes has a positive effect on a consumer’s health; however, it poses a potential risk due to their level of heavy metals in their body. Heavy metals can be naturally found in the environment, but their concentration can be increased with anthropogenic activities. Samples of tuna (Thunnus albacares) were collected at a fishery market. The potentially toxic elements (arsenic, cadmium, lead, and mercury) were determined with a validated method in the flesh of fish using inductively plasma optical emission spectrometry after microwave digestion. Generally, the average concentration of them was below the official limit values regulated by the European Union, except for lead. Based on the concentrations of arsenic (inorganic derivates: 0.05 ± 0.02 mg/kg) and cadmium (0.03 ± 0.01 mg/kg) in the tuna fish samples, and their calculated EDI values (As: 0.03–0.09 µg/kg/day; Cd: 0.05–0.07 µg/kg/day), the investigated food could be declared safe for human consumption. Generally, mercury content was below the official regulated limit, and the calculated EDI value was below the dietary reference value (0.3 μg/kg/day) in most of the samples (90%), exceeding it only in two samples (0.69 and 0.82 μg/kg/day); thus, they may not be harmful to the consumer. The concentration of lead above the official maximum limit (0.30 mg/kg) in 40% of tuna samples (0.30–1.59 mg/kg), as well as the exceeding of the dietary reference value for lead (adult: 0.16 μg/kg/day; children: 0.26 μg/kg/day) based on the calculated EDI values (0.28–1.49 μg/kg/day), draw attention to the importance of environmental pollution and the protection of consumers’ health.