CrI3, a two-dimensional layered material, has recently attracted a lot of research interest due to its exotic magnetic property. However, the synthesis of a CrI3 monolayer (ML) by vapor-phase deposition has not been achieved by current experimental endeavors, which require a better understanding of the vapor-phase growth mechanism involved. In this study, we theoretically simulate the growth of the CrI3 ML on the Si(111) surface by using a first-principles calculation. Our key finding is that an iodine buffer layer on the Si surface is crucial to the formation of the CrI3 ML by stabilizing the precursor and also reacting with the precursor. Moreover, our simulation reveals that the CrI2 cluster as the growth building unit, which can be formed by controlling the chemical potential of the I source, is preferred for the formation of CrI3 ML. We think that our work will provide insightful guidance for the experimental synthesis of CrI3 ML in the future.