Branch-and-Cut is a widely-used method for solving integer programming problems exactly. In recent years, researchers have been exploring ways to use Machine Learning to improve the decision-making process of Branch-and-Cut algorithms. While much of this research focuses on selecting nodes, variables, and cuts [11,9,26], less attention has been paid to designing efficient cut generation strategies in Branch-and-Cut algorithms, despite its large impact on the algorithm performance. In this paper, we focus on improving the generation of subtour elimination constraints, a core and compulsory class of cuts in Branch-and-Cut algorithms devoted to solving the Traveling Salesman Problem, which is one of the most studied combinatorial optimization problems. Our approach takes advantage of Machine Learning to address two questions before launching the separation routine to find cuts at a node of the search tree: 1) Do violated subtour elimination constraints exist? 2) If yes, is it worth generating them? We consider the former as a binary classification problem and adopt a Graph Neural Network as a classifier. By formulating subtour elimination constraint generation as a Markov decision problem, the latter can be handled through an agent trained by reinforcement learning. Our method can leverage the underlying graph structure of fractional solutions in the search tree to enhance its decision-making. Furthermore, once trained, the proposed Machine Learning model can be applied to any graph of any size (in terms of the number of vertices and edges). Numerical results show that our approach can significantly accelerate the performance of subtour elimination constraints in Branchand-Cut algorithms for the Traveling Salesman Problem.