We study the transient properties of a bistable kinetic system driven by correlated noises for the cases of multiplicative coloured noise and additive white noise. The mean first-passage time (MFPT) of the system is calculated. From numerical computations we find that: (i) the MFPT of the bistable system are affected by the correlation time of multiplicative coloured noise T and the cross-correlation strength between noises λ and, T and λ play the same roles in the MFPT. (ii) The MFPT corresponding to weakly correlated noises and strongly correlated noises exhibits the very different behaviour and there is a one-peak structure in the MFPT for strongly correlated noises. The peak grows highly as T increases, which means that the noisy colour causes the suppression effect of the escape rate to become more pronounced.
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