Increasing the generation of electric power from renewable energy sources (RESs) creates important challenges to transmission system operators (TSOs) for balancing the power system. To address these challenges, adequate system flexibility is required. In this context, TSOs carry out flexibility assessment studies to evaluate the flexibility level of the power system and ensure that a stable operation of the transmission system under high RESs integration can be achieved. These studies take into consideration numerous scenarios incorporating different assumptions for temperature, RESs penetration, load growth, and hydraulic conditions. Until now, flexibility studies usually solve the standard unit commitment problem and evaluate if the flexibility level is adequate. Although this approach provides quite accurate results, the computational requirements are significant, resulting in limiting the scenarios chosen for examination. In this paper, deep learning approaches are examined, and more precisely, an integrated system of two recurrent neural networks with long short-term memory cells is designed to carry out the flexibility assessment task, aiming at the reduction in the computational time required by the optimization process. The output of this neural network system is then used to calculate the probability of flexibility shortages. The proposed method is evaluated based on data from the Hellenic transmission system, providing quite promising results in (a) accurately calculating the probability of insufficient flexibility and (b) achieving a significant decrease in computational time. This novel approach could notably facilitate TSOs since more scenarios can be included, exploiting the computational efficiency of the method. In this way, a more complete evaluation of the flexibility level of the power system can be achieved and thus help to ensure the stable and reliable operation of the transmission system.