In the paper we consider first order stabilized Runge-Kutta-Chebyshev methods (RKCs) application to discrete delay differential equations (DDEs) and perform a linear stability analysis studying the standard linear test equation with real coefficients. We try two variants of RKCs extension for DDEs: the first, suitable for constant delays and constant time-steps; the second, with linear interpolation between the time-mesh points. It is shown that delay-independent stability regions are larger if using interpolation. As for ordinary differential equations RKCs have points of stability vanishing along the real values of the coefficient of the non-delayed term. We use damped RKCs to improve the stability regions and find an "optimal" damping factor to maximize the numerical stabiity region coverage of the exact stability domain. All the results are confirmed by numerical simulations.