This paper, under the excitations of correlated multiplicative and additive noises, the mean firstpassage times (MFPTs) in a bistable system with two different kinds of time delays are investigated. First, based on an approximated method, the analytical expressions of the MFPTs are obtained. Then, the effects of the multiplicative noise intensity, the delays and the noise coupling strength on the MFPTs are studied. For the positive coupling strength, the MFPT T1(x-→x+) is a nonlinear function of the multiplicative noise intensity and the two delays, but an increasing function of the noise coupling strength. Moreover, the delays contained in the determined and random forces respectively influence the maximum and the corresponding noise level on T1(x-→x+). The MFPT T2(x+→x-) is a nonlinear function of the delay that contained in the determined force, but a decreasing function of the multiplicative noise intensity, the other kind of delay and the noise coupling strength.