The objective of this research is to evaluate the ε-minimum overflow threshold of the Bayes codes for a Markov source. In the lossless variable-length source coding problem, typical criteria are the mean codeword length and the overflow probability. The overflow probability is the probability with which a codeword length per source symbol exceeds a threshold and the ε-minimum overflow threshold is defined. In the non-universal setting, the Shannon code is optimal under the mean codeword length and the ε-minimum overflow threshold for the Shannon code is derived for an i.i.d. source. On the other hand, in the universal setting, the Bayes code is one of universal codes which minimize the mean codeword length under the Bayes criterion. However, few studies have been done on the overflow probability for the Bayes codes. In this paper, we assume a stationary ergodic finite order Markov source and derive the upper and lower bounds of the ε-minimum overflow threshold of the Bayes codes.