This paper presents a training method for neural networks and the employment of MSE (mean scare error) values as the basis of a decision regarding the identity claim of a speaker in a recurrent neural networks based speaker verification system. Recurrent neural networks (RNNs) are employed to capture temporally dynamic characteristics of speech signal. In the process of supervised learning for RNNs, target outputs are automatically generated and the generated target outputs are made to represent the temporal variation of input speech sounds. To increase the capability of discriminating between the true speaker and an impostor, a discriminative training method for RNNs is presented. This paper shows the use and the effectiveness of the MSE value, which is obtained from the Euclidean distance between the target outputs and the outputs of networks for test speech sounds of a speaker, as the basis of speaker verification. In terms of equal error rates, results of experiments, which have been performed using the Korean speech database, show that the proposed speaker verification system exhibits better performance than a conventional hidden Markov model based speaker verification system.