This paper proposes a client selection method for federated learning (FL) when the computation and communication resource of clients cannot be estimated; the method trains a machine learning (ML) model using the rich data and computational resources of mobile clients without collecting their data in central systems. Conventional FL with client selection estimates the required time for an FL round from a given clients computation power and throughput and determines a client set to reduce time consumption in FL rounds. However, it is difficult to obtain accurate resource information for all clients before the FL process is conducted because the available computation and communication resources change easily based on background computation tasks, background traffic, bottleneck links, etc. Consequently, the FL operator must select clients through exploration and exploitation processes. This paper proposes a multi-armed bandit (MAB)-based client selection method to solve the exploration and exploitation trade-off and reduce the time consumption for FL in mobile networks. The proposed method balances the selection of clients for which the amount of resources is uncertain and those known to have a large amount of resources. The simulation evaluation demonstrated that the proposed scheme requires less learning time than the conventional method in the resource fluctuating scenario.