Gas turbine engines (GTEs) are highly nonlinear control-nonaffine systems. Deriving their physics-based models can be challenging, particularly when some critical parameters can be difficult to measure or determine otherwise. The possible solution to this problem lies in data-driven approaches that extract the underlying dynamical features from the measured trajectories. In this paper, a modified method based on the Koopman operator theory is utilized for GTE identification, as it allows a description of the system in a globally linear manner. Linear parameter-varying and bilinear Koopman models were considered. Extended dynamic mode decomposition was employed to obtain a finite-dimensional approximation of the operator. A novel application of genetic algorithm is proposed with a specifically crafted objective function that allows a constrained nonconvex optimization of functions in the Koopman observable subspace while forcing the solution to satisfy specified performance requirements via penalization. The prediction error was given by a numerical integration of lifted state-space equations as it reflects the usage of the model for real-time prediction in practice. Effects of parameter settings were analyzed. The prediction performance is compared to an in-house model that was validated against commercial simulation software, and the accuracy-complexity trade-off is discussed. Furthermore, model parameter corrections are introduced to cope with the effects of changing flight conditions. Finally, Koopman eigenfunctions and modes are investigated to analyze the underlying dynamics. Results point to suitability of the proposed approach for the identification of GTE dynamics and future optimal control system design.