In this paper, we propose a novel methodology based on Koopman mode analysis to predict the transient stability of a power system in real-time. The method assesses the stability of the system based on a sliding sampling window of PMU measurements, and it detects the evolving instabilities by predicting future samples and investigating the computed Koopman eigenvalues. This approach is also able to identify alarm conditions, which include slowly evolving instabilities that may not be detected by predicting future samples in a limited time horizon. Identifying these conditions provides additional time to prepare a proper set of emergency control actions to be performed when necessary. Using the proposed method, groups of coherent generators that play roles in the evolving instabilities can also be identified, contributing to the design of a defensive islanding scheme for unstable cases. The efficacy of the proposed approach is demonstrated by simulating its performance with three test systems of different scales.