Opioids such as morphine are the first choice in acute and chronic pain treatment. However, they lead to addiction. Several studies have searched (i) to find a molecule that can replace morphine use or (ii) to reduce its adverse effects. This work aimed to evaluate whether (–)-Borneol [(–)-BOR], a bicyclic monoterpene, in doses of 25, 50, and 100 mg/kg (i.p.), has an antiaddictive effect on morphine (5 mg/kg, i.p.) and reduces its withdrawal symptoms precipitated by naloxone (8 mg/kg, i.p.) in Swiss mice. Furthermore, the (–)-BOR genotoxic potential was also investigated by the comet assay. The antiaddictive effect of (–)-BOR was evaluated by the conditioned preference place (CPP). The CPP was induced by morphine administration during the conditioning phase. The effects of (–)-BOR on the rewarding characteristics of morphine were tested in mice with the administration of (–)-BOR, naloxone, or vehicle (NaCl 0.9%), 30 min before morphine. This work also investigated the (–)-BOR effect on morphine withdrawal symptoms precipitated by naloxone. Morphine withdrawal symptoms were induced by administering morphine twice daily for 5 days, precipitated by naloxone administration on the sixth day. The effect of (–)-BOR on reducing morphine withdrawal symptoms was evaluated in mice that received (–)-BOR before daily morphine administration. Finally, the comet assay was performed to assess the DNA damage degree caused by the (–)-BOR (100 mg/kg, i.p.) administration. The comet assay was performed on peripheral blood taken from the tail of each animal. Cyclophosphamide (50 mg/kg, i.p.) was used to induce DNA damage. After starting the protocol, analyses were performed for 4 h (acute effect) and 24 h (repair effect). The (–)-BOR (100 mg/kg, i.p.) significantly attenuated (*** p < 0.001) the acquisition of morphine-induced CPP and reduced only the jumping behavior in the morphine withdrawal model. The best-studied dose was 100 mg/kg, being evaluated, then, in the comet assay. (–)-BOR at 100 mg/kg did not show the genotoxic effect when compared with the cyclophosphamide group (CYCLO, 50 mg/kg, i.p.) after 4 h or 24 h, a period that corresponded to the repair time of DNA fragmentation. The study showed that (–)-BOR attenuated the acquisition of CPP by morphine and made opioid withdrawal milder. In the comet assay, although (–)-BOR caused DNA damage, this damage was significantly less than the damage by CYCLO, at either 4 h or 24 h after the treatments.