This paper concerns the study of the numerical approximation a semilinear parabolic equation subject to Neumann boundary conditions and positive initial data. We find some conditions under which the solution of a semidiscrete form of the above problem quenches in a fi- nite time and estimate its semidiscrete quenching time. We also prove that the semidiscrete quenching time converges to the real one when the mesh size goes to zero. A similar study has been also investigated taking a discrete form of the above problem. Finally, we give some numerical experiments to illustrate our analysis.