A combination of numerical and asymptotic methods was used to study how burning velocities associated with the use of an eddy-break-up-like model are affected by the presence of quenching. Quenching was introduced by setting the mean reaction rate to zero as soon as the progress variable representative of the combustion process was below some quenching value. The geometry of a one-dimensional premixed flame that propagates in a constant turbulence field was considered. The burning velocity, an eigenvalue of the problem as soon as quenching is present, is a decreasing function of the quenching value. When the quenching value tends to zero, the burning velocity and the mean structure of the reacting zone tend to the value and structure predicted by the Kolmogorov-Petrovskii-Piskunov theory. The convergence rate towards this limit is extremely slow. This indicates that, in practical calculations using an eddy-break-up-like model, the propagation characteristics of the calculated mean reacting zones can be significantly affected by the accuracy of the numerical scheme, as well as by the treatment of the boundary conditions.