This article proposes and numerically demonstrates a widely tunable on-chip Raman soliton source based on a cascaded As2Se3 waveguide. The cascaded sub-waveguides (input and output) with varying widths act as nonlinear devices, while a tapered waveguide is arranged between them to achieve low-loss transmission. The input waveguide provides anomalous dispersion in the near-infrared band, thereby enabling the 1.96 µm source for Raman soliton self-frequency shift (SSFS) pumping. The output waveguide exhibits large anomalous dispersion and good mode confinement in the mid-infrared band thus supporting further SSFS process. A 2.29∼4.57 µm tunable Raman source is theoretically realized in this on-chip platform. This work presents a simple and easy-to-implement strategy to extend the tuning range of on-chip sources. Notably, to the best of our knowledge, this is the first demonstration of the cascading strategy for SSFS process in an on-chip platform. The proposed tunable source has great potential in integrated spectroscopy, gas sensing, and LiDAR applications.