Slot filling and intent detection are two highly correlated tasks in spoken language understanding (SLU). Recent SLU research attempts to explore zero-shot prompting techniques in large language models to alleviate the data scarcity problem. Nevertheless, the existing prompting work ignores the cross-task interaction information for SLU, which leads to sub-optimal performance. To solve this problem, we present the pioneering work of Cross-task Interactive Prompting (CroPrompt) for SLU, which enables the model to interactively leverage the information exchange across the correlated tasks in SLU. Additionally, we further introduce a multi-task self-consistency mechanism to mitigate the error propagation caused by the intent information injection. We conduct extensive experiments on the standard SLU benchmark and the results reveal that CroPrompt consistently outperforms the existing prompting approaches. In addition, the multi-task self-consistency mechanism can effectively ease the error propagation issue, thereby enhancing the performance. We hope this work can inspire more research on cross-task prompting for SLU.