A numerical algorithm for the Mañé critical value approximation of eikonal Hamilton–Jacobi equations on networks is presented. The proposed method is based on the long time approximation of the corresponding evolutive problem, which is solved numerically with a semi-Lagrangian scheme. More specifically, some numerical tests in the case of networks in \( \mathbb{R}^2 \) are shown, to illustrate the convergence of the algorithm.