The maximal likelihood transition path(MLP)is informative to explain the
mechanism of the noise-induced transition, such as chemical reactions,
biological switches, nucleation processes, etc. In present paper, we
investigate the MLP between two metastable states in a delayed stochastic
system by employing a recently developed minimum action method. A modified
version of the Maier-Stein model with linear delayed feedback is considered as
an example. By an analysis using small delay approximation, we find that, as
the delay time increases, there is a threshold $\tau_c$ above which the MLP
undergos a symmetry breaking bifurcation via a transverse instability. The
bifurcation is verified by numerical simulation for the transition path within
a finite time interval $T$. Further simulation shows that $\tau_c$ is
independent of $T$, indicating that the bifurcation can occur in the
$T\rightarrow\infty$ limit. What's more, the dependence of transition
probability between metastable states on $\tau$ shows distinct difference below
and above the threshold. Our results indicate that time delay can dramatically
influence the mechanism of noise-induced transition.