In this paper, we investigate the acceptability of a non-task-oriented dialogue system that uses utterances to get closer psychologically. We defined utterances to get closer psychologically as “utterances that express intimacy with the other person or a favorable feeling toward the other person, such as joking, sympathy, compliment, or non-honorifics utterances (like a friend)”. Conventional research has reported that jokes, non-honorifics utterances, and compliments are useful for building a smooth relationship between a dialogue system and a user. On the other hand, individual differences in acceptability to utterances to get closer psychologically are considered to be large. In particular, we believe that the personality characteristics of the user affect the acceptability of utterances to get closer psychologically. So, we set research question 1: “How do user personality traits affect the acceptability of a non-task-oriented dialogue system with utterances get closer psychologically?” Also, utterances get closer psychologically has the risk of making the interlocutor uncomfortable. Therefore, in considering the implementation of utterances gets closer psychologically in a dialogue system, it is useful to examine how different strategies of utterances get closer psychologically affect the acceptability of a chatting dialogue system. So, we set research question 2: “How do different utterance strategies to get closer psychologically affect the acceptability of chatting dialogue systems?” To discuss these research questions, we conducted a dialogue experiment using a rule-based non-task-oriented dialogue system (n = 82). The results showed that for RQ1, among the five personality characteristics targeted in this experiment, the user’s diligence was related to the evaluation of the non-task-oriented dialogue system for utterance strategies to get closer psychologically used in this experiment in the subjective index, and extroversion, neurotic tendency, and openness in the objective index (likability based on user utterances). For research question 2, the experimental results showed that the acceptability between utterance strategies to get closer psychologically was significantly different in the viewpoint of the subjective index. These findings contribute to the design of a non-task-oriented dialogue system.