A deniable authentication protocol allows a sender to transfer an authenticated message to a receiver in such a way that the receiver cannot prove to a third party about the source of the message. In 2001, Deng et al. proposed two deniable authentication protocols under the communication model defined by Aumann and Rabin in 1998. In this paper, we show that these two protocols have a common issue, which incurs vulnerability to the person-in-the-middle (PIM) attack. We also propose a modification for solving the problem.