Landslide dam formation can be influenced by the erosive capacity of river flow and the dynamic characteristics of the landslide. When the deposition rate of a landslide that reaches a river is higher than the erosion rate of river flow, the landslide can form a dam by blocking the channel. Hence, in this paper, a dimensionless discharge threshold for landslide dam formation considering landslide and river dynamics is established and studied numerically. A two-layer depth-averaged model coupled with an erosion term is presented to simulate river and landslide movements and their interactions. Several numerical cases are simulated to study the influence of landslide and river dynamics on the critical threshold for dam formation by considering some key factors, such as landslide velocity and the angle between the river and landslide transport directions. Through the simulations, three types of landslide intrusion into river can be reflected: a dam forms quickly, a dam forms or does not form close to a critical state, and no dam forms. The results show that these factors together affect the process of dam formation if the difference between the landslide and river discharges is relatively small. All results are helpful to further clarify the formation of such dams for natural hazard prevention under future climate change conditions.