The effects of polymer chain flexibility and elongational flow field on the isotropic -nematic phase transition are studied by applying a freely-jointed rods model to the Onsager theory. The biphasic region becomes wider and the difference of order parameters between the coexisting phases becomes larger as the flexibility increases. Also, the onset concentration of highly ordered nematic phase becomes lower, the biphasic region narrower and a difference of the order parameters between the coexisting phases smaller as the stretching rate increases. It is shown that there exists a critical point at a sufficiently high stretching rate, which means the existence of a stable monophase above the critical point. We emphasize that there exists not only a unstable biphasic state but also a stable biphasic state in a weakly stretching rate. Thus the order parameter has double values in this stable biphasic region.