People can generate high-quality ideas by building on each other's ideas. By enabling individuals to contribute their ideas at their own comfortable time and method (i.e., asynchronous ideation), they can deeply engage in ideation and improve idea quality. However, running asynchronous ideation faces a practical constraint. Whereas trained human facilitators are needed to guide effective idea exchange, they cannot be continuously available to engage with individuals joining at varying hours. In this paper, we ask how chatbots can be designed to facilitate asynchronous ideation. For this, we adopted the guidelines found in the literature about human facilitators and designed two chatbots: one provides a structured ideation process, and another adapts the ideation process to individuals' ideation performance. We invited 48 participants to generate and select ideas by interacting with one of our chatbots and invited an expert facilitator to review our chatbots. We found that both chatbots can guide users to build on each other's ideas and converge them into a few satisfying ideas. However, we also found the chatbots' limitations in social interaction with collaborators, which only human facilitators can provide. Accordingly, we conclude that chatbots can be promising facilitators of asynchronous ideation, but hybrid facilitation with human facilitators would be needed to address the social aspects of collaborative ideation.