We present a new class of numerical methods for quasilinear first order partial functional differential equations. The numerical methods are difference schemes which are implicit with respect to the time variable. The existence of approximate solutions is proved by using a theorem on difference inequalities. The proof of the stability is based on a comparison technique with nonlinear estimates of the Perron type for given operators. Numerical experiments are presented. Results obtained in this paper can be applied to differential integral problems and to equations with deviated variables.