This work reports the effect of Ti diffusion on the bipolar resistive switching in Au/BiFeO3/Pt/Ti capacitor-like structures. Polycrystalline BiFeO3 thin films are deposited by pulsed laser deposition at different temperatures on Pt/Ti/SiO2/Si substrates. From the energy filtered transmission electron microscopy and Rutherford backscattering spectrometry it is observed that Ti diffusion occurs if the deposition temperature is above 600°C. The current-voltage (I-V) curves indicate that resistive switching can only be achieved in Au/BiFeO3/Pt/Ti capacitor-like structures where this Ti diffusion occurs. The effect of Ti diffusion is confirmed by the BiFeO3 thin films deposited on Pt/sapphire and Pt/Ti/sapphire substrates. The resistive switching needs no electroforming process, and is incorporated with rectifying properties which is potentially useful to suppress the sneak current in a crossbar architecture. Those specific features open a promising alternative concept for nonvolatile memory devices as well as for other memristive devices like synapses in neuromorphic circuits.