We present a theoretical study of high-order harmonic generation (HHG) driven by ultrashort optical pulses with different kind of chirps. The goal of the present work is to perform a detailed study to clarify the relevant parameters in the chirped pulses to achieve a noticeable cut-off extensions in HHG. These chirped pulses are generated using both linear and nonlinear dispersive media. The description of the physical mechanisms origin responsible for this extension is, however, not usually reported with enough detail in the literature. The study of the behaviour of the harmonic cut-off with this kind of pulses is carried out in the classical context, by the integration of the Newton-Lorentz equation complemented with the quantum approach, based on the integration of the time dependent Schrödinger equation in full dimensions (TDSE-3D).