Multilevel resistive switching in memristive devices is vital for applications in non-volatile memory and neuromorphic computing. In this study, we report on the multilevel resistive switching characteristics in SnSe/SrTiO3(STO) heterojunction-based memory devices with silver (Ag) and copper (Cu) top electrodes. The SnSe/STO-based memory devices present bipolar resistive switching (RS) with two orders of magnitude on/off ratio, which is reliable and stable. Moreover, multilevel state switching is achieved in the devices by sweeping voltage with current compliance to SET the device from high resistance state (HRS) to low resistance state (LRS) and RESET from LRS to HRS by voltage pulses without compliance current. With Ag and Cu top electrodes, respectively, eight and six levels of resistance switching were demonstrated in the SnSe/SrTiO3 heterostructures with a Pt bottom electrode. These results suggest that a SnSe/STO heterojunction-based memristor is promising for applications in neuromorphic computing as a synaptic device.