Existing recommendation systems have focused on two paradigms: 1- historical user-item interaction-based recommendations and 2- conversational recommendations. Conversational recommendation systems facilitate natural language dialogues between users and the system, allowing the system to solicit users' explicit needs while enabling users to inquire about recommendations and provide feedback. Due to substantial advancements in natural language processing, conversational recommendation systems have gained prominence. Existing conversational recommendation datasets have greatly facilitated research in their respective domains. Despite the exponential growth in mobile users and apps in recent years, research in conversational mobile app recommender systems has faced substantial constraints. This limitation can primarily be attributed to the lack of high-quality benchmark datasets specifically tailored for mobile apps. To facilitate research for conversational mobile app recommendations, we introduce MobileConvRec. MobileConvRec simulates conversations by leveraging real user interactions with mobile apps on the Google Play store, originally captured in large-scale mobile app recommendation dataset MobileRec. The proposed conversational recommendation dataset synergizes sequential user-item interactions, which reflect implicit user preferences, with comprehensive multi-turn conversations to effectively grasp explicit user needs. MobileConvRec consists of over 12K multi-turn recommendation-related conversations spanning 45 app categories. Moreover, MobileConvRec presents rich metadata for each app such as permissions data, security and privacy-related information, and binary executables of apps, among others. We demonstrate that MobileConvRec can serve as an excellent testbed for conversational mobile app recommendation through a comparative study of several pre-trained large language models.