Extreme-ultraviolet pulses can propagate through ionised solid-density targets, unlike optical pulses and, thus, have the potential to probe the interior of such plasmas on sub-femtosecond timescales. We present a synthetic diagnostic method for solid-density laser-generated plasmas based on the dispersion of an extreme-ultraviolet attosecond probe pulse, in a pump-probe scheme. We demonstrate the theoretical feasibility of this approach through calculating the dispersion of an extreme-ultraviolet probe pulse propagating through a laser-generated plasma. The plasma dynamics is calculated using a particle-in-cell simulation, whereas the dispersion of the probe is calculated with an external pseudo-spectral wave solver, allowing for high accuracy when calculating the dispersion. The application of this method is illustrated on thin-film plastic and aluminium targets irradiated by a high-intensity pump pulse. By comparing the dispersion of the probe pulse at different delays relative to the pump pulse, it is possible to follow the evolution of the plasma as it disintegrates. The high-frequency end of the dispersion provides information on the line-integrated electron density, whereas lower frequencies are more affected by the highest density encountered along the path of the probe. In addition, the presence of thin-film interference could be used to study the evolution of the plasma surface.