The radio access network (RAN) will be an integral part of the sixth generation (6G) of mobile networks. By using several advanced technologies (e.g., virtualization, cloud, and edge computing), it aims to address the stringent networking and computing requirements of new applications and offer high quality of service and experience levels to the consumers. However, the optimal allocation of computing and radio resources can be chal-lenging due to the heterogeneity of the network and the stringent constraints imposed by the new application requirements. This work is focused on leveraging swarm intelligence methods in an Open RAN to offer a zero-touch management network archi-tecture that autonomously orchestrates its resources taking into account several constraints. Specifically, three swarm intelligence methods are evaluated and compared, namely the Grey Wolf Optimizer, the Salp Swarm Algorithm, and the Particle Swarm Optimization. The results show that the Grey Wolf Optimizer features the best performance in solving the joint offloading and resource allocation problem in edge computing scenarios.