The present study aims to explore how second language (L2) speakers process four types of presupposition triggers in an online self-paced reading task and an offline acceptability judgment task. The four types of triggers are definite expressions with "the," the factive verb "know," the change-of-state verb "stop" and the additive particle "also." Of particular interest is L2 speakers' ability to accommodate a presupposition when it is neither falsified nor supported in the context. The results showed that the L2 speakers accommodated presuppositions in a native-like manner in the offline judgment task, and that accommodation was taxing and less automatic in online processing. Additionally, advanced L2 speakers' interpretations were affected by the types of presupposition triggers. We discuss the results in connection to the Interface Hypothesis and hope to contribute to a more precise understanding of information integration in L2 processing, as well as L2 acquisition at the semantics--pragmatics interface.